This first segment of dialog happens the first time you talk to them after beating the Magus in this era, so you can see it much earlier.

Fiona: My husband Marco had gone
as a soldier to fight the Magus's army,
army, but he has returned to me...
I, I...

Marco: Relax.
From now on, we'll be together for good.

MARCO: Don't worry.
I'm home for good.

There had, at one point, been a small forest around the villa, but now there are only a few lone trees. The sidequest appears as soon as the party tells the woman with the seedling in 12,000 B.C. to save it, as early as meeting her for the first time in Zeal Palace. Once that's done, an odd vortex forms in the desert nearby.

Note that this isn't missable. She'll turn up in the Lingering Village Town Square after the Dalton incident, so you can go there and tell her to save it any time you want—even if you previously ignored her, or even told her to go ahead and burn it.

As this is the only one of these sidequests that does not require flight capabilities, it can be completed relatively early.

Fiona: No matter what I plant,
it soon withers...
It's the work of monsters that have
settled in somewhere in the desert.
If I were to plant this seedling with mysterious
powers that was passed down for generations,
it could even restore this desert to green earth...
But I can't plant it with monsters nesting
in the desert cave...

Marco: It seems the monsters nesting in
the desert cave are weak against water.
But with my strength, trying to beat those
monsters would be...

The entrance resembles a whirlpool of sand. Inside is a cave system, where "Strange Happening" plays and the sand moves in irregular currents.

The Laracargo (Hexapod) and Sandrago (Mohavor) appear here, often arriving suddenly and attacking without warning. Both resist Aether and Fire attacks, and the Laracargo has high defense, while the Sandrago has low defense but high evasion, so Water or Dark magic or combo attacks tend to work best. Anything Water will also make them more vulnerable.

Things get more interesting deeper inside when the ground starts shaking.

A strange skeletal something sticks partway out of the ground, then sinks back under the sands.

Marle: It must be that monster's fault
the forest is dying out so fast!

{Marle}: So that's the monster
responsible for this desert!

Lucca: It looks like that monster is
the source of the desertization.

{Lucca}: The monsters are creating a
desert here.

Frog: Th-that thing must be what turned
the forest into desert!

{Frog}: So, THEY changeth the forest
into the desert!

Robo: The progressing desertization
must BE that monster's fault.

{Robo}: The monsters are causing
the desertification here.

Magus: Melfikk...
So that's behind the desertization...

Two 'k's make it sound twice as sinister.

Marle: But a monster that moves
around underground...?
You can't even guess where to run into it!

{Marle}: But how can one find a
monster that roams the inner earth?

Lucca: But a monster that moves
underground is a nuisance.

{Lucca}: It will be hard to defeat a
monster that travels underground!

Frog: But being up against a thing
that moves around underground...
This guy's a pain.

{Frog}: How to slayeth monsters
within the earth...
'Tis a very difficult task.

Robo: HOWEVER, a monster that can
come and go underground...

{Robo}: But monsters that travel
underground are impossible to find!

Ayla: Bones moving around underground!
Big trouble to find!

{Ayla}: Hard to find monster in
earth!

Catching the monster when it's above ground starts combat.

Elemental attacks inflict no damage to any part of Melfikk (Retinite), and all but Water heal the core. Though they inflict no damage, Water attacks will stiffen the sandy bones, dropping the defense of the torso (nameless monster, ID 182) and legs (nameless monster, ID 181). Other attacks will restore their defense, but only if the core is dead. To make the battle easier, keep the core alive by healing it with any element except Water to counteract the health it loses when the torso and legs drain from it. This will also limit the kinds of attacks the boss can use. Steal from the core for a Speed Capsule.

Marle: Now the trees won't die off any
more either. Miss Fiona will be glad.

Lucca: Now the cause of the desertization
is removed. We ought to tell miss Fiona.

{Lucca}: We've neutralized the
source of this desert.
We should tell Fiona.

Frog: Now there's nothing left to worry about.
Let's let Fiona know.

Robo: The progress of the desertization
SHOULD now SUBSIDE.
LET US notify miss Fiona.

Ayla: Bones that ruined forest dead.
Fiona glad!

Marco: I can't believe you exterminated
those monsters for us!
How can we ever thank you...?

MARCO: You killed the monsters?!
How can I ever thank you?

Fiona: Thank you for exterminating
the monsters for us.
I will try doing all I can once more.
But to make the desert into forest
once again, we will need an
unimaginably long time...
Ahh...
If only there were someone who could
keep working for hundreds of years, we
could turn this desert back to forest...

Robo: ......
Well, EVERYONE?
I shall remain HERE and act
as miss Fiona's HELPER.
You can just COLLECT ME after
THIS desert IS revived as forest.
May I remain HERE and act as
HELPER?

Robo goes out into the desert, and can from now on be seen performing various tasks related to growing the forest.

The area, which had been almost barren, is now covered with forest, and this unfamiliar building stands in the middle. There are several nuns, priestesses, or whatever term best applies, inside.

This is a bit of a loose translation. 神体 (shintai) refers to an object in which the spirit of a deity resides, and the line suggests that they consider Robo to be an inanimate object that was animated by a god.

She sells various hats with protection from status ailments, including the Safeguard Hat. They're expensive, but are some of the strongest headgear, and anyone can wear them. Robo sits inert on a pedestal in the back, looking worn down and rusty. They move him to the ground and he starts moving.

Robo: WH-WHERE...?
OH...
OHH...
{char1}...
I... I MISSED YOU...
NO... to YOU, it WAS a matter
of an INSTANT.
HOWEVER, for ME, 400 years
WAS a long time...
HOWEVER, HARDSHIPS bear results,
and the forest HAS BEEN revived.
......
NOW, tonight, SHALL WE NOT
celebrate our 400-year REUNION?

That night, everyone relaxes around a campfire in the forest. "The Farthest Reaches of Time" plays. Lucca continues repairing Robo as he speaks.

Robo: Having now taken a journey of
400 years, I HAVE realized something.
WE HAD thought that the Gates appeared
due to the distortions from Lavos's power,
but I HAVE come to the conclusion that
we may have been mistaken.

{Robo}: After 400 years of
experience, I have come to think that
Lavos may not be responsible for the
Gates.

Robo: I cannot be CONFIDENT about it,
but could they not BE the work of someone
who wanted to show us something...?
Something in multiple eras spanned by Gates.
Alternately, MAYBE that someone wanted
to see this something personally.
As though to think back upon the way
their own life went...

{Robo}: I have come to think that
someone, or something wanted us to
see all this.
The different events over time, that
we have witnessed.
It is almost as if some entity wanted
to relive its past.

Ayla: Ayla understand that.
Legend say people see all
memories up to then when die!

{Ayla}: {Ayla} know!
When people die, elders say, see
whole life pass by!

Frog: They say when you die, the memories
etched deeply into your heart while you
lived come to mind one after another.
Some are happy memories, yes, but
generally they're sorrowful memories.

{Frog}: 'Tis true that mortals do
relive their most profound memories
before death claimeth them.
Yet those memories most often are
sad ones.

Robo: Such strong feelings as
"I want to go back to that time" or
"if I had done things differently then"
...ARE sure to call memories to mind.

{Robo}: Thinking things like, "If only
I had done this," or, "I shouldn't
have done that..." triggers
unpleasant, old memories.

Marle: Do you have one, Lucca?
A moment you want to return to?

{Marle}: Is there a point in time you'd
want to return to, {Lucca}?

Lucca abruptly stops working and stands, looking at the ground as she replies.

{Marle}: I'm sorry, was that
something I shouldn't have asked?

Lucca: I try to think about it as
little as possible.
After all, it just wears me out to.

Lucca resumes working.

Frog: Anyway...
whoever's behind these memories
is really tangled up with Lavos.
Every one of these eras involves it.

{Frog}: Lavos playeth an integral
role in the fortunes of this Entity...

Magus: ......
Well, who's this someone, then?

Robo: I do NOT know whose
memories they are.
Perhaps not a person...
MAYBE a larger existence.
The day we understand that MAY
come when OUR journey ends.
...SHALL we call it a night?

Who is this mysterious entity? The going fan theory seems to be the planet itself, which fits quite well given what Lavos is doing to it.

The music fades and everyone goes to sleep. Lucca, however, wakes up in the middle of the night. There's a faint background noise similar to the sound of travel through a Gate. Investigating, Lucca finds a dormant Gate in the nearby forest and activates it, but it's red when opens instead of the usual blue. After hesitating briefly, she enters. The sounds get louder as the Gate deposits her into a room that resembles hers, but lacks the usual clutter of books, machines, and chemicals. She fiddles with her glasses as if agitated.

There's a note on the ground.

"A.D. 990 6/24
Dad promised to go hiking, but he
skipped out on it to do research.
Lucca doesn't get it at all...
But that's fine. Lucca's gonna be a
normal bride, so she doesn't gotta
know about science and stuff!

This appears to be from a diary written by Lucca ten years in the past. The third person is an indication of childishness. The date is in sensible year-month-day order, as opposed to the what-were-they-thinking month-day-year order hardly used anywhere but the USA, to be honest.

The kitchen downstairs has another note on the table.

"The passcode to stop the machine is...
my most beloved person.
Enter this in an emergency.
　　　　　　　　　　　　　　　　Taban"

As soon as Lucca enters the main room of the house, a scene starts and the sounds get even louder. Lucca's mother Lara is in the room along with a young purple-haired girl and a large machine. The others don't notice Lucca, who looks shocked.

Lara: Dear me, what's this machine...?
Taban says not to get close to it because
it's dangerous, but he's gotten it so dirty.

She walks to one side of the contraption, where it looks like a conveyor belt, then seems surprised.

Lara: Huh?
The hem of my skirt got caught in it.
Lucca, oh, Lucca, give me a hand.

The little girl, who must be the Lucca of ten years past, goes over to help.

With a clicking sound, the machine starts to move, dragging Lara in.

Lara: !!
Lucca! Stop the machine!
Enter the passcode.

Lucca: I don't get it!
Mom!

Young Lucca hurries to the controls, but just runs back and forth helplessly. However, the older Lucca is in a position to change the past...

Lara: Ahh, it's no good...
Darling!!

The scene fades out, the sounds stop, and there's a scream. Fade in to (older) Lucca collapsed at the foot of her bed. There are now two notes in the room, blocking every exit, except for the Gate, which has reappeared.

"A.D. 990 7/1
I heard dad and the
doctor talking.
But I wish I hadn't...
Mom... her legs...

"A.D. 990 7/15
If only I understood machines better,
that never would have happened...
If only I...

Note that this entry is written in first person. The shift indicates a movement from childish innocence to adult maturity.

The Gate returns Lucca to A.D. 1000. Robo stands in the clearing. Everyone else is still sleeping.

Robo: I WILL give her MY legs.
They ARE well-balanced ultra-
miniaturized bipedal walking parts.
If you would just attach caterpillars
or something similar to ME...
PLEASE do not be depressed.

{Robo}: I'll give you my legs!
They're lightweight and well
balanced.
You could mount some treads on me
instead.

Lucca: Thanks.
I have such a good friend...

{Lucca}: Thanks, you're such a good
friend.

Robo: Friend...
I, a robot...

{Robo}: A...friend?
Me, a robot?
You called me your friend?

The code in the Japanese version is L A L A, not L A R A. While this is one of several possible ways to romanize ララ, it's questionable as a name in English, so they changed it for the NA release. It's also possible that the ambiguity of how to romanize her name was done deliberately to make it more difficult to get the code correct. Text dumps from a prerelease version suggest that, at that point, it used a code that was outright impossible to enter.

The ambient sounds fade. Young Lucca runs up to her mother.

Fade in to Lucca collapsed at the foot of her bed. There are now two notes in the room, blocking every exit except for the Gate, which has reappeared, though one of the notes is just the one that was here when Lucca arrived.

"A.D. 990 7/2
Wow, thank goodness. I'm thinking I'll learn
a little more about machines so I'll be okay
in case something like that happens again.

Since there are (in Japanese) neither any first-person pronouns nor Lucca's name, it's unclear whether this entry is in first or third person. This may mean she grew up a little, but not as abruptly as in the scenario with the tragedy.

The Gate returns Lucca to A.D. 1000. Robo stands in the clearing. Everyone else is still sleeping.

Robo: It HAS BEEN weighing
on your mind the whole time.
You ARE a kind person, Lucca.

The scene ends the same way regardless of which branch was taken. Robo slides open some of his plating, revealing something shiny inside.

Robo: I will give you a present.
I MADE it by hardening the RESIN
of trees that grew in the forest.
This jewel HAS four hundred years of meaning.
Please make some use of it.

Lucca: Thanks...
Robo.

{Lucca}: {Robo}...
you're so sweet.

Thus concludes the sidequest. Rewards: Speed Capsule (steal), Verdant Dream, various other treasure, the hat shop, and the infamous "Entity" discussion.

Saving Lara's legs or not seems like more of a personal matter than anything that could have any lasting effect on history, but that's only a side trip. The real impact here was in revitalizing a forest that had died out completely in the original timeline. As for as what effect it might have to inspire what appears to be some kind of religious order that regards Robo as a god... let's leave that question alone.