"Would you mind ___ing?" is a good way to ask politely for something that isn't too much trouble for the other person. Some more examples:

Would you mind giving me a hand with this?

Would you mind covering for me while I'm out?

Would you mind forwarding me his reply?

In all of these examples, you're asking the other person to do something. If you want to ask for permission to do something yourself, say "Would you mind if I ___?"

A: Are you ready to go?

B: Actually, would you mind if I stay here and rest instead?