Mine is not actually one from the books directly. In the introduction to "deleted scenes and chapters from Dune" in The Road to Dune, we have a most reasonable outlook:
Many details are inconsistent with the published versions, and these scenes should be considered drafts, not "canon".
Perfectly sensible. When you find something in FH's notes, you probably shouldn't consider it canon anyway because it's a not, not published Dune material, but at least they recognise that if it contradicts published material, you call it a draft.
In the FAQ in the DN site, we have this little fart:
Q: In the preludes Jessica is almost forced on Duke Leto. Yet in the DUNE classic, we read that the Duke actually sent buyers to purchase her.
A: The Leto/Jessica meeting scene in HOUSE HARKONNEN was actually written by Frank Herbert himself and found in his notes. The events shown in HOUSE HARKONNEN are consistent with the original notes.
And of course, FH & KJA insist that their novels are canon. So they completely, irrefutably, inarguably contradicted themselves.
It's one thing to make a blunder (we can see FH did a few himself) but I think this mistake gets elevated to a new level because one is on the official website explaining "seeming" inconsistencies, and the other is in possibly the only genuinely good thing BH has done for the Duniverse: an official presentation of his notes. It goes beyond a mere error for me - they make a rule and they blatantly break it.
Of course, the most annoying thing that has been done as regards inconsistencies is the whole "FH's novels are a distorted history" thing, but that is not an inconsistency itsself, just an extremely insulting and flimsy retcon.
So, any of you have a favourite inconsistency?
Some other candidates:
Mass-amnesia about Teg having super-speed. I mean seriously guys, how'd you manage that one?
Dune-Piter and prelude-Piter being different gholas, despite him saying in Dune that he was the one that predicted a daughter for the Atreides, and having fear in his voice when the Baron was angry about it.
It being apparently common knowledge that the Tleilaxu were deeply religious, despite that only being discovered by Taraza and Odrade thousands of years later.