Papa Wheelies (papa-wheelies.com) in Portsmouth NH rents road bikes. I do business with the shop, but don't have any experience with their rentals. There's nice riding along the coast of NH out of Portsmouth, so you might want to spend a half day (or more) on your way up to enjoy it.

Shops in Portland, ME advertise rentals as well. You'll find a lot of rentals in Bar Harbor, but that's a bit further up the coast.