I like the 4-player & Battle mini-games the most. Here's what I remember:
Mario Party 5:
Dinger Derby (Good at this. Got the maximum record.

)
Leaf Leap (I'm pretty good at this. =D)
Later Skater (moar racing

)
Tug-o-War (was it called by that?)
Cage in Cooking
Mario Party 6:
Snow Brawl (Snowfight! I like it! Maybe this could've worked on 2-on-2 or 4-player better...)
Seer Terror (Bowser - Funny as always.)
What Goes Up... (The night version. It's a lot easier.)
Snow Whirled (takes kinda skill...)
Cannonball Fun (reminds me of 4-player split screen FPS-games...)
Rocky Road (I always tease my partners by hitting them constantly xD)
Wrasslin Rapids (Free For All!)
Control Schtick (just you wait 'til you get very far, it's so fast you'll laugh yourself out!)
Tally Me Banana (maybe it's just the water that makes good...)
Mario Party 7:
Kart Wheeled (Kart racing! I love it! ^_^ But I always get the worst starting place... -_-)
The Final Countdown (dedudeeduu... dedudeeduude- wait... Oh yeah! I like how the platforms keep disappearing. >8D)
Hip Hop Drop
Apes of Wrath (very exciting at times...)
Pretty much every 8-Player mini-game because they are just unique and awesome.
Ice Moves (^_^)