One cute idea is to use a onesie as the invitation. Use fabric paint on the front to write "______ is 1!" Then put all the party details on a large 1 and pin it to the back of the onesie with a diaper pin.
Any theme is good, the child is only 1 year old! Use a generic theme such as balloons and streamers and party hats for the invitations, that way they can be reused in later years too.