What does “Sleep through the Second Coming” mean?

• The 'Second Coming' is expected to be a big deal.

• For someone to sleep through such a big deal gives an idea of their character, that they are 'blissfully unaware' of major things, namely the problem that Kane himself is so concerned with.

That is all.

For people who fervently believe in "salvation thro Jesus", the 2nd Coming of Jesus is the big kahuna that would vindicate their steadfast faith in their religion. According to their theology, when (and imo if ever) Jesus returns, even all those who had died would be awaken (or already awaken in the preceding rapture). Dispensationalism, tribulation and millennialism not withstanding.

So sleeping thro the 2nd Coming is a humourous expression among Fundamentalist Christians about someone who is so tired that would not be aroused even by the 2nd Coming.

To tell a story in this manner, with references to Christian theology is to put in perspective the religious environment in which the characters in the story exist.

It may not be common in general literature, but it is an immediately understood humourous expression among Fundamentalist Christians worldwide.

If you had somehow gotten yourself into the midst of a Fundamentalist Christian group like sitting with your friend at lunch and then incidentally all her/his fundamentalist friends joined us for lunch, you would be bewildered by all the private jokes they banter at each other, all based on verses of the Bible or concepts of their doctrines.

An example is

She eats like a pre-millennialist in a post-millennial event. She would think she still had another thousand years to finish her lunch.

This phrase is a simple hyperbole.

According to theology, the Second Coming (of Christ, that is) is an event that includes lot of loud sound, flashes of light, commotion, and activity and it will affect everyone (for specifics, see the Signs of Christ's return on the Second Coming wikipedia page). It is an event of such magnitude that it should wake everyone one up.

Saying that she would sleep through it is just a fanciful way of saying that she is an exceptionally deep sleeper, and is not awakened by activity in the environment. Saying someone "can sleep through X" is a common expression (especially "can sleep through a storm").

This point is even further pronounced by the juxtaposition with the prior sentence in the passage:

A curtain only had to flutter in the breeze and he would wake up. She would probably sleep through the Second Coming.

The first sentence "A curtain only had to flutter..." is also a hyperbole, meant to say that he is an exceptionally light sleeper.

The passage says that she is "blissfully unaware of his latest problem" because by comparison to being sleep deprived, being able to sleep is blissful, and that she is not aware that he is unable to sleep. Without other supporting context, the expression doesn't imply anything at all about her character in general.