I have multiple Select buttons next to player's names in order for a user to choose which player they want in a game. A user can't just click on a player's name because that gives additional stats, which is why there needs to be a select button. They are rather ugly but I also don't want checkboxes. If anyone can think of something more appealing to use I'm open to hear it.
Not a direct answer - but have a look at the way many of the arcade video games handled multiplayer selection.
There are lots of screenshots online - here's one:
Or, better yet - go down to the video arcade and see how they work.
Why do this?
Because arcade video games are in the harshest commercial market - anything that degrades the user experience leads directly to lost revenue because the player goes to another game machine and plays there instead.
Of course, you'll need to do some adaptation - your players are unlikely to have four joysticks attached to the machine, but video games don't usually have keyboards, and so on.
If possible, I would use just one button called "Continue", or "Go", or "Play".
This button would be beneath the entire list. Clicking on a player displays the stats and highlights that player and then once the button is pushed, the selection is made.
This avoids repeating the button and provides a natural flow - when I find a player with the stats I like, I stop looking and press the action button to select.
If that's not possible, consider whether viewing the stats is the main action in this screen?
Will most of your users be just viewing stats most of the time?
It sounds like after viewing the stats once or twice, their main focus on the page would be selecting the player.
I would give the user a 'token' (small circle with "user", "P1", etc... inside) that selects their player and the user drags/drops on the player they would like to use.
This allows them to select the player they want while still viewing that player's stats and without cluttering up the screen with more checkboxes or buttons.
You can have a "highlight" of some sort follow the mouse. Imaginge this:
 2 3 4 5 6
The numbers are your character images. When you move the mouse the bracket follows it. Clicking your mouse 'selects' that character. You'll have to figure out another way to communicate that a particular character has been selected.
We could use more information about how the screen is organized. Assuming it's split (horizontally/vertically) between the list of characters and a stats panel - why not place one Select button in the stats panel? You could click through the list of characters, the stats would update according to current selection, and the Select button would stay in one place.
You could display the information when the mouse hovers over and select on click. Another way would be to display information on single click and select on double click.