re: Any cheap(ish) ways to refinish granite counters?

None of the shite at a big box stores is going to help you. I just repolished my marble countertops myself. Marble is way softer than granite, so you're going to have your work cut out.

Your options:

1. Go from a polished finish to a honed or satin finish.

2. Spend forever trying to achieve a machine polish.

3. Pay someone to do it, but be warned it's almost as expensive as just replacing the stone.

If you aren't in love with what you have, I'd forego 1-3 and change your counters.

If you are going to do it yourself, you will need a decent variable speed polisher and some diamond polishing pads.

Diamond Polishing Pads 4" inch Wet/Dry Set of 11+1 Backer Pad Best Value Granite Concrete LINK