This is a tough one to answer, because Dudley's name changes with each revamp of continuity.
In the Fawcett era, I honestly don't know if his full name was ever given, but it was revealed "Dudley" was his last name, as he had a niece named Freckles Dudley.
It has been fairly well established by E Nelson Bridwell (and accepted by this fan) that on the live action TV series, the character of Mentor was actually Dudley (although very early developmental notes on the production reveal Mentor was supposed to be Shazam himself, or perhaps Billy/Capt Marvel's predicessor, a la Teth Adam). Since he was often referred to, on the show, as Mr Mentor, perhaps his full name was Dudley Mentor?
In Roy Thomas' 1980s revamp/false start, Dudley was made Billy's real paternal uncle, thus his name being Dudley Batson.
In Jerry Ordway's restart (my all-time least favorite version of Dudley: an alcoholic janitor), he was given the "clever" name of Dudley H. Dudley.
As for the current continuity of BILLY BATSON & THE MAGIC OF SHAZAM, Dudley has only made one brief cameo to date as a hot dog vendor in the Jeff Smith mini-series.
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