"To remember the name" means to memorize or keep in mind someone's name so that you can recall it later when needed. It implies being able to recognize and recall the specific name of a person.
Sentences with «to remember the name»
I'm sorry I do not remember the name of the Doctor or the vitamins, That has been 36 years ago. (downshiftology.com)
My husband request this one by name and he can't even remember the name of our kids! (icantbelieveitsnotbutter.com)
Just as writing things down when you were in grade school helped you remember it later, it works just as well for remembering names. (creditdonkey.com)