Although the verb nombrar exists, it is used not for giving someone a name but rather for naming (appointing) someone to a post. Le nombraron como su representante. They chose him to be their representative.
Ponerle un nombre
If you want to talk about naming a baby, the verb you want is poner.
- ¿Cómo le vamos a poner? What are you going to name her? Le vamos a poner Fabiola. We’re going to name her Fabiola. You might not have expected cómo in this question but it sounds perfectly natural.
- Se puso un nombre falso. He gave himself a fake name. As you can see from this example, poner works pronominally for naming yourself.
Fuera CFE sign, Mexico City, 2016
You can use poner not only for naming people but also for naming things:
- A la red de mi casa le puse: Mis Vecinos Apestan. I named my wifi network: Mis Vecinos Apestan.
Recall that for nicknames we often use decir: Me dicen Lalo. My friends call me Lalo.
Ponerle Jorge al niño
A cutesy and safe-for-mixed-company euphemism for having sex is ponerle Jorge al niño, literally, name the baby Jorge. Perhaps ponerle Jorge al niño works as an albur because ponerle con in general means to have sex with someone: Prométeme que no le vas a poner con ningún italiano, cosa. Promise me you won’t get it on with any Italians. (movie dialogue)