What goes into a great Mario Party-style game, which should be called a Mario Party-like, or a MP-like? This meditation will bring forth some basic ideas and then wrap it up.
Basic Ideas
There are a few factors to consider: layout, movement, rewards and pitfalls, and minigames.
First, we will consider layout. The layout is a board containing differing styles of squares and organized in what is a hopefully pleasing manner. There's reward spots and pitfall spots. In the end, it should befit the player's desires and entertain, as well as challenge. It should be sufficiently large so as to keep them busy, but not too large so as to be boring as it is "watered down" with meaningless squares. Finally, it should have a theme. Any of the boards, for example, from Super Mario Jamboree are great examples.
Next, we will consider movement. The movement should be semi- or entirely chaotic. By this, I mean typically through dice, but there can be other mechanisms, such as drawing cards from a deck of some kind and moving in a way mandated by the card.
Next, we will consider rewards and pitfalls. There should be spots rewarding players with a token of some kind. This could be coins or stars, such as in the Mario Party games, or could easily be any sort of token thematic to the MP-like. Also, there should be pitfalls. These could be traps or loss of tokens. In either case, these should manipulate the amount of tokens a player has.
Finally, we will discuss minigames. These need to be fun, quick, and plentiful. How players are presented will depends on how the MP-like desires. Mario Party, for example, will randomly select one from a presented list.
Starting A Game
So, you have your board and you have your way for movement chosen. What's next? Well, each MP-like might have other elements of play to enhance it. Some, for example, may add a bit of role-playing. Finally, all players need to agree on the number of turns. Even or odd? It really doesn't matter as long as fun is to be had! So, enjoy your endeavours and boards and games and fun!
Wrapping It Up
Keep it in mind: these are only suggestions. If you can come up with a formula in the spirit of Mario Party and it works for you, make it work!