Trying to apply for a mail in rebate does the name have to match the billing address?

I'm trying to send in a mail in rebate. The item was paid for by someone else, but shipped to me, both the billing and shipping addresses are the same just the first name is different. Does it matter which name I use to submit the rebate? Both the shipping and billing addresses are included on the packing slip that is required to send in so is it alright if I use my name because the person that bought it isn't going to be around until after the rebate is due.

Thank you