My iPod mini's battery dies within about 30 seconds. What's up with that?

That's a pretty good sign that the battery is completely shot and in need of replacing. Not suprising since it's an iPod Mini and those have been out for a number of years now.

You can choose to have Apple replace the battery for a flat fee, send it to a third party service to have them do the job for a little less, or for even less you can buy a new battery for it and swap it out yourself.

http://www.apple.com/support/ipod/service/prices/

B-rock