Going over each option:
Richard - possible. It has precedence. William the Conqueror, Geoffrey of Anjou and Henry II named their second sons after themselves.
William - also possible. Richard's elder brother was named William, Henry the Young King's short-lived son was supposedly named William, and all of them were descended from William the Conqueror.
Henry - slight chance. Even John abandoned his father at the end, but IOTL he still named his eldest son and heir Henry.
Geoffrey - I doubt it. I'm not sure how close Richard was with his brother, Geoffrey. Hell, I'm not sure Richard was close to anyone except his mother.
Edward - No way in hell. Edward was an Anglo-Saxon name and the early Angevins/Plantagenets were as far away from being Anglo-Saxon as you could get.
Alexander - Maybe, if the boy was born on crusade, and Richard was in the mood. Greco-Roman history might have been part of his education at his mother's court.
That, or something else, like Jordan (if the boy was born near or baptised in water from the Jordan river, like one of HRE Frederick II's legitimate sons. Apparently, that was one of the reasons why the name was adopted in Europe - crusaders bringing back water from the Jordan river to baptize their children.)