If you keg it, it will end up carbonated at least to some extent (unless you use some sort of wine setup using argon or nitrogen). For me, that's a dealbreaker for a sports drink that I want to drink a lot of rather quickly.
If I were going to make one to keep on tap, then I'd just start with a soda recipe that is slightly on the sweet-tart side (cola or fruit-based) and add some NaCl, KCl and Mag Sulfate (or Mag oxide). The sweetness and acidity will help mask the mild saltiness from the added salts. Just don't make it too puckering or sweet to affect the refreshing aspect.
NaCl I'd get from kosher salt. KCl I'd get from a salt substitute or bulk supplement product (something with no fillers - sold either in capsule or loose powder form). The magnesium I'd get from a supplement as well. For the amounts, I'd try to match the Sodium, Potassium and Magnesium values in Pedialyte (Gatorade actually uses too much, Pedialyte more closely matches an appropriate rehydration solution).