If you are already baptized and being received into full communion, you may pick anyone to be your sponsor as long as they meet the requirements (fully initiated Catholic, living a life consistent with the Catholic faith, over 16 but not your parent). I would discourage asking your girl friend for the same reasons everyone already has, but I know that spouses are often the sponsor for someone being received.
If you are not baptized, your parish may opt to assign you a sponsor for the period of the catechumenate. Some parishes even prefer that such a sponsor be of the same gender. That period can be quite intense and having someone who is not so close to you and more objective can be much better.
But later, when you are about to be baptized, another person of your choice, who meets all the above requirements, can be chosen to be your ‘godparent’. That person steps in to accompany you at the Rite of Election, at the celebration of initiation and during the period of mystagogy. There is nothing that forbids your girlfriend from becoming your godmother, but you may want to think of someone else you’d like to have in that role.