I have tons of picks in this category. First that comes to mind is Cyberspeed (PS). It's not an extremely good game (I gave it a 6.2 in my review), but I liked it for various reasons. For one thing, it was the first Playstation game I ever played, so it holds that special place from a nostalgia standpoint. Also, the wacky TV commercials in it are pretty funny, and were totally new at the time.
Another one is the second Playstation game I ever played, Blazing Dragons. I love the story, the characters, the overall goofiness of the game. It's so fun from that standpoint. But, the gameplay is nothing special, and overall if I had to grade it harshly, I probably wouldn't give it a very high score. Still, one of my favorite games of all time.
Also, I dare to say, Resident Evil: Survivor. For those who haven't played this, I envy you. The first thing you'll notice, is that the voice acting is bad by RESIDENT EVIL STANDARDS. I'm dead serious. The gameplay is a bad light gun game without the light gun. The graphics were horrible for the time, the story is sub par, the characters and bland and uninteresting. Overall, easily the worst game in the Resident Evil franchise. But, for some reason, I liked it. I guess with me, it falls into the "So bad, it's good" category. It's pretty funny too. I still pop it in on occasion, just to torture myself.
Am I done? Nope. Staying on the Playstation, ECW Hardcore Revolution. Sure, it's pretty much an exact clone of Acclaim's previous WWF games, and it doesn't do the slightest bit of justice to the source material. But, the intro video is freakin' awesome, Joey Stiles does a good job on commentary, and overall the gameplay is okay. I still play it once in a blue moon.
Moving to the PS2. The Scorpion King: Rise of the Akkadian. I liked the movie, and I was a huge fan of The Rock at the time. It was just a beat-em-up where you play as The Rock and beat the crap out of everything in sight. For this reason, I liked it. But, it sucked. The gameplay was a bland button masher, the AI was stupid, the sound was garbage, the game was filled with small glitches and annoyances. Overall, it wasn't very good. I found it to be fun, though.
Finally, 24: The Game. I'm such a huge fan of the show, that I simply HAD to play the game. Also, I got it essentially for free because I got it during a Buy 1 Get 1 Free Preowned sale at Blockbuster with Smackdown vs. Raw 2006 a while back. The gameplay was fun, but glitchy and pretty unremarkable. There was some nice variety, but none of the games within The Game really stood out on their own. The story was awesome, but it had its flaws as well. Overall, I loved it though.
In conclusion, my standards for video games are probably way too low. But, I like it that way. It allows me to enjoy games I normally wouldn't. Anyway, do have any similar feelings toward certain games?