Hi! You can use both... If the gender of the cat is specific you can use her/his. If it's not you can use its

You can use either one. People tend to use "her" if it is a pet, if they feel affectionate toward it, and they know the cat's gender. In your case, you have given the cat a name, and it is even a female name. So I would guess that you think of your cat as female. If so you would say "I have a cat, her name is Mia and she is grey." If you are thinking of the cat as "an animal," neither male nor female, you would say "I have a cat, its name is Mia and it is grey." Either way, the two pronouns should agree: both feminine ("her"/"she") or both neutral ("its"/"it.")

Her

its

I have a grey cat named Mia.