I am a 13-letter word, _H_T_ _ _I_ _ME_. Fishermen like me, doctors hate me, kids love to eat me. What am I?
The answer given is "chathuringmes", alleged to be the scientific name for a worm (i.e. used as bait / disliked as a parasite / and eaten by children as the "jelly worm" sweet). The trouble is, there's no reliable verification whatsoever for this term. Why and how the yarn got started is pretty murky. The best suggestion I can find, via Answerbag, is this Times of India piece from 2006 - You said SMS? How puzzling! - that refers to a whole genre of insoluble/hoax riddle-format questions circulating via SMS. Another example cited is:
Your left side is your right side, Your right side is your wrong side. The right side, for your back side is the front side. What is it?
Not that the idea is new. Nik Kershaw's pop song The Riddle attracted a deal of speculation as to the meaning of
Near a tree by a river
There's a hole in the ground
Where an old man of Aran
Goes around and around
As Wikipedia explains:
The "riddling" lyrics caused much puzzlement and speculation among listeners as to their meaning, which was further fuelled by record company MCA's decision to run a competition to work out the meaning - without either telling Nik or even bothering to ask him what the meaning was. Had they done so, they would have discovered that there is in fact no meaning at all - the lyrics were nothing but gibberish thrown together to fit the music, or in Nik's own words: "nonsense, rubbish, bollocks, the confused ramblings of an 80s popstar".
See also The Straight Dope for an even older example: Lewis Carroll's Why is a raven like a writing desk? which, as originally invented, had no answer. (I have to admire Sam Loyd's solution: "Because Poe wrote on both").