You probably know this now since you asked this a month ago but you got me researching it, and others might be curious, so I'll post here:
French distinguishes between the comparative forms of "good"(bon) and "well" (bien). Mieux = more well / plus bien, meilleur = more good / plus bon. So you might say "he swims well": "il nage bien" and "he swims better": "il nage mieux"
Yes, when I realized this technical difference (adjective vs adverb) I was able to remember much better. I was having trouble remembering whether mieux, for instance, went with bien or with bon, but I think the easiest way to remember is that bon and meilleur are adjectives and thus must agree with gender (bonne/meilleure), while bien and mieux are adverbs and are immutable.