Outside of the obvious franchises of Final Fantasy, Mega Man, Dragon Warrior/Quest, Castlevania and licensed games which don't really count anyway like the TMNT games...
Blaster Master had a decent sequel on PS1
Metal gear spun into something a bit more epic
Splatterhouse sequel/remake as pretty good (genesis series that went to ps3)
Chrono Cross is a sequel to Chrono Trigger (obviously)
Some various other RPG sequels such as Breath of Fire,
Clock Tower had sequels on PS1 and PS2, Haunting Ground is also a spiritual successor
Contra continues on, but if you're a fan of the originals, you probably don't want to bother with the disc based sequels
STAY AWAY FROM BIONIC COMMANDO! trust me...