Why not kill two birds with one stone by giving the weak underpowered characters the ability to hard-counter the currently "overpowered" sub-set of characters with skills or passive skills added onto their currently existing skills and passives that directly effect the currently "Overpowered" characters.
Vlad OP?: Give one of the UP characters a massive damage boost with an AoE skill against Vlad when hes in Sanguine Pool, (and only Vlad when hes in Sanguine Pool.)
Kassadin, Cho'Gath and Malzahar OP?: Give one of the weak UP characters a passive skill based temporary immunity that protects against "void silencing" effects once every 15/14/13/12/11 minutes.
Tanks OP?: Give one of the weak UP carries a Magic Resistance based skill that deals damage based on HP% with a lowering mana cost with each level you put in it.