This interesting little term comes from a long-running column by John Cameron Andrieu Bingham Michael Morton, though he preferred to be called J. B. Morton. Who wouldn't, with a handle like that? Anyway, Morton wrote his column for the Daily Express from 1924 to 1975; it was called By the Way, and Morton wrote it under the pen name "Beachcomber." One of the characters in the column was a detective by the name of Prodnose -- a man who prodded his nose into other people's business.
The name became a noun and a verb:
"Lee was a prodnose who prodnosed into the conversation."
Sadly, prodnosing is an equal-opportunity, gender neutral activity.