The name of that girl
The name of that girl's
What do you mean?
The name of that girl is correct!!!
You can use the other sentence this way:
That girlīs name.
So when to put and "'s" after people to express possession relationship?
The name of that girl's dog is Rex.
As MT noted, "That girl's name ..."
That girl's boyfriend is over there.
That girl's bag is brown.
That boy's motorcycle is a Honda.