Fiona's Sidequest - 600 AD Middle Ages
The first side quest that the old man in the End of Time mentions is, "In the middle ages, a woman's sheer determination brings a forest back to life." This quote is about Fiona's Villa in 600 AD the Middle Ages, you passed right by this area earlier in the game but at that point in time, Fiona's Husband had yet to return.
At Fiona's Villa, speak with her and her husband and they'll tell you about an area to the south which has appeared called Sunken Desert. They ask you to go to this dungeon and clear it of all the monsters that you find there. When I went through this area I was in my high 30s with my team - not much was very challenging. The boss in this cave, Retinite, you'll want to use Ice/Water spells on to weaken him to physical attacks. Also, defeat his upper body parts first.
Once the dungeon is cleared of monsters, put Robo in your party and go back to talk to Fiona. Robo will offer to stay and help Fiona, accept his offer and then get into the Epoch and fast forward to 1000 AD. Return to the same location that Fiona's Villa was at and you'll see that the area has changed quite a bit.
Inside of Fiona's Shrine there'll be three nuns, two of them will tell you about Robo's remains which are found at the northern most point in this building. One of the nuns will offer to sell you four different helms, Sight Cap (Prevents Chaos), Memory Cap (Prevents Lock), Time Hat (Prevents Stop and Slow) and Vigil Hat which gives protection against status ailments.
If you can afford some of these helms I recommend purchasing them. Especially the Vigil Hat, great defense and protection against status ailments is always good. After you're done with the nuns, inspect Robo's remains in the northern most portion of the church to trigger a scene.
After you gain control of Lucca, you'll find that in the eastern portion of the area there's a time portal, hop into it and you'll find yourself in Lucca's House when she was a kid. Another short scene follows of Lucca's mother getting trapped in the conveyor belt. In order to turn off the machine and save Lucca's mother you'll need to input the correct password.
Depending on which version of this game you're playing, the password might be a little bit different. I've included a list of all the passwords that I could find on the internet for each of the different Chrono Trigger remakes.
SNES/Nintendo DS Code: L A R A (Lucca's mother's name)
Ps1: L1, Circle, R1, Circle
After you save Lucca's mother (or don't) return to the time portal and back to your team. If you successfully saved Lucca's mother then return to her house in 1000 AD and speak with Taban who is inside. He'll give you two pieces of armor for Lucca which are best in slot items for her. Taban Helm (Magic Defense +10) and Taban Suit (Speed +3 and Fire Resist Increased).
That's all there is to this side quest!