There's only two that I have ever played that I enjoyed, but one of them may not count. The one that may not count is the fishing in Zelda OoT. I enjoy it because depending on the size of your catch, you get really good rewards. A heart piece as a kid, and the gold scale as an adult. It also gives you a distraction from the main game if you're kinda stressing out. Getting Epona is another fun one.
The other minigame that I really enjoy is Triple Triad in FF8. It's a simple card game that most always took for granted, until they taught Quezacotl the card mod ability. Then they saw that the cards could be modified into really useful and important items. Also I just think it's loads of fun to play, and collect the cards. I just never go anywhere near the Queen Of Cards side quest. That always goes so far beyond enraging me, that I'll not only blow a gasket, everyone reading this will get a migraine that they can't find a reason for. When there's something like that, that pisses me off that much, I quit right then and there. It's also the reason I don't play DKC anymore. That game went above and beyond the psycho bitch factor. I have all 3 on SNES, but I don't touch them. That's a blown valve that I don't need.