You have answered your own question - the 'most logical' badge is the one that you are closest to achieving, or in other words the one you have progressed the closest to achieving.
One way I have noticed is if you calculate what % completion you are at for each of the badges. If there are 2 or more the same, then it is sorted by priority (gold, then silver, then bronze).
For example, one of mine:
Looking at the top row:
Fanatic and Proofreader are both at 77/100 or 77% complete, Fanatic is first because it is gold; Proofreader is second. Then in third place is Archaeologist, at 58/100 or 58% complete, finally for this example in fourth place is Marshall at 283/500 or 56.6% complete.
Observing the changes in my 'Select your next badge' screen tends to indicate to me that the % badge completion is the major part of the logic.