What's the meaning of "mean" in "in the mean time"?

As I understand it "in the mean time" means "in the time between now & a specific future occurrence." What's the meaning of "mean" here?

I assume it has something to do with "average" but it's not clear to me exactly what.

The mean in meantime and meanwhile does, as you suspect, come from the same root as the word meaning mathematical average.

The original meaning of this mean is middle, and, with respect to meantime, it branches off to meaning intermediate and then further off to mean¹:

Intermediate in time; coming or occurring between two points of time or two events; intervening

The mathematical meaning also comes from this middle meaning, but branches off into a meaning of average, middling or moderate, which is then co-opted by maths.

1. "mean, adj.2." OED Online. Oxford University Press, December 2014. Web. 26 January 2015.

Meantime / Meanwhile « The Word Detective

The “mean” in “meantime” and “meanwhile” is the adjective “mean” meaning “occurring between two points in time,” based on the noun “mean,” middle point, from the Latin “medianus,” in the middle. (This is a separate word from the other adjective “mean” in the sense of “nasty,” which comes from Germanic roots meaning “common, low-quality.” And the verb “to mean” in the sense of “to connote” or “to intend” comes from yet other roots meaning “to tell or say.”) The “while” in “meanwhile” is the noun form of the word, meaning “a period of time,” thus serving the same function as the “time” in “meantime.” We also use “while,” of course, as an adverb meaning “at the same time” (“While the cat’s away, the mice will file a restraining order”).