There's not one perfect answer for every situation. If you don't hear the name the 1st time, just say that. Or, if the caller just gives his/her 1st name, state that you need the last name as well, and the reason why.
I encounter this often. Although I am a native English speaker, most of my callers speak Spanish. So, I am confronted with two problems: First, the need to know the full name of the caller; and second, the fact that I am not trained in the culturally-appropiate way to ask my client's identity in Spanish. (And the most appropriate way to ask this, may vary from country to country, anyway.)
My advice is to listen to the caller, and to respond in a way that seems correct for that situation.
Your tone of voice, and your attitude, probably are more important than the exact words that you use.