'Sniveling' is being nervous or suppressing tears. 'Ley' I would think just means lay. I suspect JKR used this word because of its mythological connections, such as in 'ley lines' (lines joining ancient monuments or religious sites) and because it sounds like lay. (It is a homophone).

In original: "Dud ley snieveled". It's just misprint - excessive whitespace. Correct phrase: "Dudley snieveled".