This page explains how to do the Sunken Desert sidequest.
How to Start the Quest
For this quest, go to Fiona's place (north of Porre) in 600 A.D. You should see a swirling hole near her house. If not, swing by 12,000 B.C., the Commons, and talk to the sapling lady and tell her to plant it. Then head back to Fiona's place and the swirling hole should be there.
Go to the Sunken Desert
Switch your party to Frog, Robo, and Ayla. Equip them with Shadow-absorbing armor, as well as status-protecting gear.
Check on the swirling hole to go in, and let yourself get drawn into the center.
Don't worry about the chests in here for now, there are monsters about who will go away when you beat the boss, making it easier to open the chests.
Since the desert here is slippery, you have to run if you want to get anywhere. So, run above the chest near you, then just below the stairs leading up. Go straight down and you'll have to fight. Use Water first, then physical attacks. To avoid more monsters, head diagonally down-left, then right and down the stairs.
In this next area, each time the floor shakes you lose 30 HP, so quickly go around and find the boss who keeps rearing his ugly head.
Charm a Speed Tab from the Retinite Core. First use Water 2 to weaken everything, then Beast Toss the top half. If the core dies, then the bottom and top's defense will go up with each hit. You can heal the core with non-water magic attacks to make sure this doesn't happen. Once you kill the top, go for the bottom with Spin Kick. If you kill the bottom before the core, quickly kill the core with physical attacks or it'll run away and you miss out on some extra experience.
Now go around the room and get the seven chests. Then head up to the first room.
There are four chests here and a Power Tab a little left of the center of the room.
Put Robo in your party, go to Fiona's and talk to her. Agree to let Robo stay behind. You'll see him working on the soil outside.
Go to 1000 A.D. and now there's a big forest there.
SAVE FIRST. There's a part next that you might mess up, and if you don't save first, you won't be able to try again.
Go into the center structure, Fiona's Shrine, and talk to the nuns if you want to buy some useful Helms. IF YOU DID NOT SAVE, DO IT NOW. Then go up and check on that big, Robo-shaped hunk of metal to reactivate Robo.
A long scene follows. When Lucca awakens, exit southeast. Go through the gate.
Head down the stairs, then down through the door below. Then when the machine turns on, walk all the way down and all the way right, press A, then when it says "Enter the password," press a button to dismiss the dialog. You should hear a chime. Then press L, A, R, A (Lucca's mom's name, which is on a piece of paper in a room above). If you don't do it in time, then Lucca's mother will be caught in the machine and history won't be changed.
Lucca will have an emotional moment, then you can read the papers on the floor (which somehow block your passage out of the room), and then the only place you can go is through the gate.
Go down and talk to Robo to get the GreenDream.
If you stopped the machine, then when you go to Lucca's house next time, her mom is now able to walk around.