Chwazi is great (and free), but you do have to have everyone get their finger on the device's screen (and, btw, some older devices can't recognize more than 5 fingers).
Another free option that just requires you to select the player-count and then tells you who goes first (i.e. no fingers needed) is Who Goes First? [0]
But the most streamlined app is Fast First [1] for $1.99 which, every time you open it, generates images of players sitting around tables for player counts ranging from 2-7 (for Android and it looks like up to 17 with iOS, depending on the device) with a start-player selected for each player-count. That is, you get a screenful of images representing a table of 2 players, a table of 3 players, etc. each with its own random start-player selected. So, if you have, e.g., 5 players at your table, you just open the app and look at the image for 5 players and use the start-player shown in that image.
It does take away the fun interactivity of the finger-based apps, but, that makes it fast to use.
