There are 16 4-player minigames in Mario Party 4, many of which focus around rapidly pressing a button. Only one of the 16 is a coin-collecting bonus game, where everyone can earn coins.
Manta Rings
Rules: Dive through the rings while swimming in the ocean. Each ring is worth one point. If any enemies hit you, you'll lose points.
Controls: 
: Move
: Press repeatedly to go forward
Advice: The gold Manta rings are worth three points, so try to grab as many gold ones as you can!
The best strategy in this game is to go after chains of rings. Frequently, there will be several rings more or less in front of each other, worth 3 or 4 points. Stay at the back, dodging enemies until a chain of rings and/or a gold ring appears. When there aren't any chains, simply go after the rings that the other players ignored. If a second player stays at the back of the screen with you, swim slightly ahead of them so that you can get to the rings before them.