The mother of my son baby is in the hospital who have the right to take care of the baby while she in the hospital her mother or him?

You need to provide more info about the relationships, custody issues etc before we can answer this.

In addition to what Scottgem said, I'm also wondering if you are asking who has a legal right or who has a "moral" right to watch the baby.

I'm going to put my neck out here and say the one to look after the baby is the one the baby's mother trusts the most.