According to a similar question in The Guardian's Notes and Queries
http://www.guardian.co.uk/notesandqueries/query/0,
5753,-14156,00.html
the monarch is apparently allowed to choose which name they like. (However, the reasons behind their choice are not always clear, although it is rumoured that Charles may use a different name since the two previous King Charles's didn't do particularly well.)
Incidentally, Edward VIII's real name wasn't David, it was actually Edward Albert Christian George Andrew Patrick David - so although he was known as David, he used his real name when he became King. For other real names of monarchs, see Debrett's
http://www.debretts.co.uk/royal_connections/kings_
and_queens_of_britain.html