Sun Stone Sidequest - Starts at Sun Palace in 2300 AD
The Sun Stone sidequest is the fourth one that the Old Man in The End of Time tells you about. The flavor text for this quest is, "And there's a very special stone that can shine its light on each generation, from the distant past to the far future."
To get started you'll need to head on over to Sun Palace in 2300 AD. It's recommended that you bring Lucca and/or other people in your party who are highly resistant to Flame attacks. The first boss you fight inside this palace, Son of Sun, will attack you with powerful fire attacks and having Flame Resistant armor will basically negate all of them.
Once you've defeated the Son of Sun you'll want to grab the Moon Stone in this dungeon before leaving. With the Moon Stone in hand travel back to 65,000,000 BC and fly the Epoch to the small island west of Dactyl's Nest (shown in my map location below) where Sun Keep is.
Place the Moon Stone at this location and then travel to 1000 AD and return to the Sun Keep. You'll notice that the Moon Stone is gone - a party member of yours will comment that it might have been stolen from someone in this time period. Exit the Sun Keep and travel over to Porre Village to the east. You'll notice when you arrive that one of the houses here is sparkling.
Now you have to do a bunch of running around between time periods. Here's a simple list of what you have to do and in which order:
1) 1000 AD Porre Village: Buy Jerky for 9900G from Snail Stop.
2) 600 AD Porre Village: Give Jerky to housewife in the Elder's House - do NOT sell it to her!! Also inspect the Sealed Chests in this house but don't open them; this will power them up and double our loot.
3) 1000 AD Porre Village: Go to the Mayor's house and speak to him after giving the Jerky to the person in 600 AD. He'll be a nice person now and will give you the Sun Stone. Grab the Sealed Chests from the second floor to get the White Mail and Black Mail.
4) 600 AD Porre Village: Return and collect the Sealed Chests for White Vest and Black Vest.
5) 1000 AD Sun Keep: Place the Moon Stone in the Sun Keep to charge it up.
Now we can finally go back to the Sun Keep in 2300 AD and inspect the Moon Stone; before you do put Lucca in your party. The Moon Stone has transformed into the Sun Stone and Lucca will get excited about how much power it wields. There'll be a few scenes before you end up at Lucca's House, she'll craft The Wonder Shot which is her ultimate weapon and her father Taban will give you Sun Shades, a great head armor piece.
There's one last thing to do too! If you've completed the Rainbow Shell Sidequest you'll be able to take the Sun Stone to Melchoir in Guardia Castle who will forge Prism Specs out of it as well as Crono's Ultimate Weapon, Rainbow!