Problem: Luke inserts Cainan into his genealogy

Luke's genealogy of Christ (Luke 3:23-38) contains an interesting error. Between Shem and Abraham, he is clearly following the genealogy in Genesis 11:10-26. There is one key exception. At one point, Luke adds a name not present in Genesis 11, namely Cainan. This is Luke 3:34-36:

Why does this occur? One explanation suggests itself: Luke was not using the original Hebrew Bible, but rather a Greek translation of it known as the Septuagint or LXX. And the Septuagint does contain Cainan's name (although see the Apologetics Press response, below).

In other words, by using the Septuagint's corrupt translation of the original genealogy in Genesis, Luke may have allowed an error to creep into into his own genealogy.

Responses

The Apologetics Press argues that, actually, mention of Cainan in the Septuagint was inserted after Luke's gospel was written, by over-zealous scribes eager to make the Septuagint concur with Luke.

If that's correct, then it's impossible that Luke's genealogy was corrupted by using the LXX as his source. We then have no obvious explanation as to where Luke got the name from. I suppose divine inspiration could be invoked.

Instead, The Apologetics Press suggests the presence of "Cainan" in Luke is itself a scribal error. Indeed, two important early manuscripts of Luke don't include the name, but the rest do, and so all modern Bibles contain it. The NET Bible's notes argue that the absence of "Cainan" in those two manuscripts of Luke may have been the result of scribes dropping it to be consistent with the original Hebrew.

Summary of positions

This whole debate is a mess. I'll summarise the possible positions as I see them:

(Luke used the Septuagint) The Hebrew didn't mention "Cainan", but the LXX did. Luke copied the LXX, introducing an error into his gospel. Someone altered Luke to agree with the Hebrew, but this version of Luke never became widespread.
(Luke made a mistake) Luke accidentally added Cainan into his genealogy, introducing an error into his gospel. Someone altered Luke to agree with the Hebrew, but this version of Luke never became widespread. Scribes then updated the LXX to agree with Luke, and this updated LXX became widespread.
(Corrupted Hebrew) The original Hebrew did indeed mention "Cainan", but the text was later corrupted. The LXX gives the original reading, and Luke is therefore correct. Someone altered Luke to agree with the corrupt Hebrew, but this version of Luke never became widespread. No correct Hebrew manuscript survived.
(Deliberate omission + inspired Luke) The Hebrew didn't mention "Cainan", and neither did the LXX. Luke - guided by inspiration - added "Cainan", who had been intentionally omitted in Genesis. Scribes then updated the LXX to agree with Luke, and this updated LXX became widespread. Someone altered Luke to agree with the Hebrew, but this version of Luke never became widespread.
(Multiple later alterations) The Hebrew didn't mention "Cainan", and neither did the LXX. Luke didn't mention "Cainan" either, but scribes copying Luke added "Cainan" by accident, and then other scribes updated the LXX to agree with the corrupted Luke. Both the corrupted Luke and the updated LXX became widespread.

Did the original LXX contain "Cainan"?

A key issue is whether the Septuagint available to Luke actually included Cainan's name or not. If it did, it's obvious where he got the name and why his genealogy differs from the Hebrew version of Genesis. I would need to be an expert in Septuagint manuscripts to decide this issue for myself, so I've decided not to classify this problem for now.