While it would take some intelligent itemization, I'd also love to see equipment that modifies specific abilities again too, as others have mentioned. This could be an interesting way to make pieces with less desirable stats (Speed for most classes, or Determination for BLM) more interesting. As a physical DPS, would you consider a Det/Speed piece over a Crit/Det one if the former improved an ability you use frequently (such as reducing the cooldown on Jump for DRG)?

While the answers would inevitably be mathed out for every particular piece, it would still add some much needed depth to the game's gearing.