How do I change the owner's name on my Mac?

That's a good question, and certainly one that I can see is going to be important to you! :-)

As far as I can tell, the main place to change the system name is in the Sharing section of System Preferences. If you look at that area (by the way, you can get to "System Preferences" off the Apple menu) you'll see the key field on the top of the window:

Change that and close the window and you should change the computer name.

There's a second area, however, that you need to address, and that's the name of the default logged in user: it's quite possible that is your ex's name. Not good. So in this case, go back into "System Preferences" and choose "Accounts" this time:

You'll want to set yourself up as an admin account if you don't already have that, and you can also click on the "Login Options" and change it so that it's your account that's the default login account when you start up the computer. If you just have a regular user account, I would suggest you set up a second account that's an admin account.

Anyway, between these two areas, you should be able to purge your computer of their identity and finally have it all to yourself!