I've got three big ones that I'd like to see.
Robot Alchemic Drive, or R.A.D. as the game case had it called. Great game from Square for the PS2. You picked one of three giant robots, and controlled said robot remotely in a style that played a bit like Fight Night. Fight giant monsters, try to avoid wrecking the city too much. Whole thing was great. Lot of the story played out like some of the early cheesy giant robot anime, which worked pretty nicely really. Would love to see the game remade and updated with bigger cities, more robots, more enemies, whole shebang.
Azure Dreams. PS1 game that was somewhere between pokemon and a strategy rpg. You had to climb this tower that randomized each time you entered, trying to gain treasures and accomplish quests for the townsfolk. Trick was that each time you left, your character was reset to level one, however you could gain a zoo worth of monsters from the tower, mostly in the form of eggs that you would hatch, that you could train to help and protect you. The monsters would never lose any experience from exiting the tower, so you would have to rely on them to really push forward in the tower. Game featured some light dating sim mechanics, and some light town building gameplay. Had a psuedo-sequel on the DS, but that game featured an unchanging tower, which kind of killed a lot of what made the original so much fun.
Tecmo's Deception. Actually more of a series, with the PS1's Deception's 1, 2, and 3 as well as the game Trapt for the PS2. Hard to really describe what genre these games were in. They were almost like puzzlers of a sort, where you played as a young girl (Deception 1 actually had you playing as a prince) who sets a variety of traps about a room in a castle type setting, then lures intruders into said traps where you will trigger them. Bonus points could be gained from building combos by doing things like making one trap send the victim flying into another, or into natural hazards built into the room. If they do remake it or make another one in the series, I'd love to see a return to the first game's freedom of customization, where you could edit and build onto your castle, and bringing back the monster making system would also be pretty nice, as it would give players something other than relying on traps alone to defeat their enemies.