Honestly one of the most overlooked elements FFIII brought to the franchise is the ability to have characters re-target enemies if the one they originally targeted gets killed. Dragon Quest held onto that mechanic for freaking ever as did a few other early RPGs well into the 16-bit era but here's FFIII being a trendsetter.

Likewise, I feel that FFVII's greatest contribution to the series was finally creating a run button instead of leaving it as skill.