Suppose you take the strings 'a' and 'z' and list all the strings that come between them in alphabetical order: ['a','b','c' ... 'x','y','z']. Take the midpoint of this list and you find 'm'. So this is kind of like taking an average of those two strings.
You could extend it to strings with more than one character, for example the midpoint between 'aa' and 'zz' would be found in the middle of the list ['aa', 'ab', 'ac' ... 'zx', 'zy', 'zz'].
Might there be a Python method somewhere that does this? If not, even knowing the name of the algorithm would help.
I began making my own routine that simply goes through both strings and finds midpoint of the first differing letter, which seemed to work great in that 'aa' and 'az' midpoint was 'am', but then it fails on 'cat', 'doggie' midpoint which it thinks is 'c'. I tried Googling for "binary search string midpoint" etc. but without knowing the name of what I am trying to do here I had little luck. 
I added my own solution as an answer
It sounds like what you want, is to treat alphabetical characters as a base-26 value between 0 and 1.  When you have strings of different length (an example in base 10), say 305 and 4202, your coming out with a midpoint of 3, since you're looking at the characters one at a time.  Instead, treat them as a floating point mantissa:  0.305 and 0.4202.  From that, it's easy to come up with a midpoint of .3626 (you can round if you'd like).
Do the same with base 26 (a=0...z=25, ba=26, bb=27, etc.) to do the calculations for letters:
cat becomes 'a.cat' and doggie becomes 'a.doggie', doing the math gives cat a decimal value of 0.078004096, doggie a value of 0.136390697, with an average of 0.107197397 which in base 26 is roughly "cumcqo"
Average of two strings in alphabetical/lexicographical order