Let's talk about games that you gave up on or lost interest in, but liked after picking them up again.
When I first got Pokemon Sapphire, I stopped playing since I just couldn't beat Norman. I decided to give it another go after buying Omega Ruby. While Hoenn is still my least favorite region, I found the game a lot more enjoyable the second time around, especially with the features they added from X and Y.
Resident Evil was another game I just couldn't get into. Granted I was 10 when I first played it, so survival horror games were foreign to me. The puzzles were too frustrating, and I was pretty bad at conserving ammo. After playing it again when I was a little older, I appreciated the games a lot more, and conservation of resources actually became my favorite thing about it.
I mentioned this before in another thread, but when I played Majora's Mask for the first time, I gave up after Woodfall Temple. It didn't help that I was really sick at the time, so for awhile, I'd associate the game with it. I could care less about the side-quests, too. For me, Zelda was all about exploring and finding secret items. Well, I caved in and bought the 3DS version, and now I'm hellbent on collecting all the masks and collecting stray fairies.