What is the name of the anesthesia that is used in the PRP hair treatment?

Anesthesia is not required for the treatment.  The research performed and the anecdotal evidence would suggest that
thickness of hair and density can improve with PRP injections.  The
number of sessions as well as the quality of PRP obtained will vary
considerably physician to physician.Kenneth Hughes, MDLos Angeles, CA

In my opinion,  most people perform PRP without anesthesia.  Some practitioners will use a  topical  anesthetic such as emla or  lidocaine. Some may use a "ring block"  using  Lidocaine with or without epinephrine. Some may put a small amount of Lidocaine into the PRP. Find a PRP expert. Ask to see photos.
PRPHAIRMD.COM 

PRP treatment does not always include an anesthetic.  Some may use topical or an injection of lidocaine or something similar.  Others may use cooling systems as we do at our center.  We believe the more concentrated the PRP the better so we prefer not to add the additional "volume" of anesthesia to the tissues.  Our patients tolerate this very well.