In the general sense, Fingersplash (my game) is about typing fast. Now that skill is a competitive skill in nature, so I'd make a lot of sense to let people compare it by battling each other in multiplayer. Hence, the multiplayer version of the game would be a versus style game in which the players were pitted against each other.
In the original game, the gamescreen is depicting a surfer struggling through waves in the rhythm of the typing of the player. In the multiplayer version, the game would be split screen. Note that even though it is split screen two computers are still needed. Now the mechanics should be that the opposing players surfer will fall if you type faster than the other player, and vice-versa. Words would still appear in the same way and the core concept of typing is of course still the same.
In the single-player game, a lot of what makes the game fun is upgrades to the player such as longer error margin, new venues, new visual upgrades etc. Then the basic concept can extend to a campaign which makes it replayable. However this progression can be taken somewhat out of the multiplayer as the competitive element is what makes it playable. Now highscores and leaderboards should be what make the game addictive.
The graphics can be more crude in the multiplayer, and maybe even more toony than originally intended (the original though was more of pixelated art). The graphics should not take any attention from the competitive element. A good comparison of how the graphics could look like is looking at the popular "Tetris Battle" from Facebook: http://www.ergame.com/uploads/big/dtFbGKNRss/2012/08/image-5019950c1b551.jpg
That's it for my multiplayer idea. Shoot me some comments if you like!