I rarely use a mime. But when I do, I have found them most effective at mimicing Calculated spells. For instance: if you miss a spell like Dont Move or Dont act, or Stop or Slow... your mime will cast the exact same spell on the same targets, effectively doubling your chances of affecting the targets. If you bring your party into a battle with some status-stopping equipment like 108 Gems and Thief Hats, maybe Black Costumes or Ribbons and Barrettes, you can totally paralyze every enemy or just let loose some well-calculated Flares and Holies...
The Mime also makes a good thief if stealing equipment is your aim. Surround an immobile enemy with thieves and mimes and you'll strip him bare in no time!