You're right to avoid "unintentional lie", because it's an oxymoron.
1. A false statement deliberately presented as being true; a falsehood.
2. Something meant to deceive or give a wrong impression.
As you can see, without the intent to deceive, it can't be called a "lie".
"Falsehood" technically sounds like it could be the neutral word you want, but most people use it as a synonym of "lie".
1. An untrue statement; a lie.
2. The practice of lying.
3. Lack of conformity to truth or fact; inaccuracy.
OK, so inaccuracy is pretty good: the information is wrong, but the conveyor of the information isn't being accused of anything. A similar word would be untruth.
If what you want is a verb, it's a bit more difficult. You're pretty much stuck rephrasing:
Jim told Sue that [...], but he was mistaken.
Jim unintentionally gave Sue false information.