The names “Elihu”, “Eliab” and “Eliel” are all used to refer to the same ancestor of Samuel in 1 Chronicles 6 and 1 Samuel 1. So the names seem to be interchangeable.

That means that the missing son is either dead or disowned.

The “missing” 8th son is Elihu:

1Chr 27:18 Of Judah, Elihu, one of the brethren of David: of Issachar, Omri the son of Michael:

Since Elihu is a “ruler” he would not be listed under Jesse/David’s lineage in 1 Chr 2:13; he would have his own (which is not listed in 1 or 2 Chronicles because these deal specifically with the dynasty of David).

A video of the answer is here (where I learned it):

http://www.youtube.com/user/TheScripturesAlone#p/a/u/0/VeudbjCENwA

Martin Richling is the teacher:
http://www.thescripturesalonebibleschool.net

I’m not convinced that we can harmonise the passages by saying that one of the brothers died between the accounts. The list in 1 Chronicles seems to be intended to be an exhaustive list, rather than tied to a particular point in time. Even if one of the brothers had died, I’d expect him to be included there.

Another possible defense is that Jesse might have disowned one of his sons between the Samuel and Chronicles passage. This might do better at explaining the positions in the birth order that you find in Chronicles.

Either way, I think the charge of error can be handled here.

It looks like the author of Chronicles has altered things in order to make David seventh, a traditionally “special” number. The NIV Study Bible basically says as much.

On the other hand, I’ve heard the suggestion that one of David’s brothers died some time after the events of 1 Samuel 16:10-11, and so David could thereafter be considered the seventh. I suppose this is possible.