it has (one's) name on it
1. It is reserved or intended for one. Are you sure you don't want a popsicle? There's a cherry one here, and it has your name on it!
2. It is a perfect match for someone's interests or tastes. Susie loves spooky things, so yeah, definitely leave that mystery novel here for her. It has her name on it.
It has someone's name on it.
Fig. It belongs to someone.; It is meant for someone. A: Is that an extra piece of cake? B: Yes, and it has your name on it.