The issue is that (e.g.) the arithmetic mean would be very influenced by the stats that gave randomly bad results - the very results you are trying to exclude by doing the test multiple times and taking an average.
From the article:
The standard error of HyperLogLog is 1.04/sqrt(m), where “m” is the number of registers used. Redis uses 16384 registers, so the standard error is 0.81%.
My understanding is that the error is very small because you're averaging over 16,000 registers.
Edit: also, I think you want the mean because it'll give you back a precise/fractional power of 2 that is pretty close to the actual count. The median would give you the nearest integer power of two, which is probably not a good estimate of the real count.
Double edit: looks like there's some research around improving averaging by throwing away outliers: http://blog.notdot.net/2012/09/Dam-Cool-Algorithms-Cardinali...