More to the point, does the value of the enchant have anything to do with the number of hits the weapon can make before discharging? I always use grand soul gems to craft and then to recharge the enchant on weapons. It seems that if I use enchanting potions to create higher value fire or frost damage, that the soul gems discharge faster --- is that true?

I don't know the formula, but the value of the enchant does factor into the number of charges, with a higher value resulting in fewer charges. The numbers are fixed when you enchant the item. The soul gems used for recharging don't change this, so you can recharge with anything you've got handy. Several Petty or Common Soul Gems is as good as one Grand. You can also top off with minor gems before heading into an area so you don't run out in mid-fight.

One thing that's very helpful is gear enchanted to reduce the cost of Destruction Spells. It also reduces the charge cost of destruction type enchantments (fire, frost, shock), so it's beneficial even if you're not a spellcaster. I've got a fighter type character with -80% Destruction cost worth of item enchantments. Charges last a loooooong time.

Enchanted Gear is effected by Enchanting Level and Perks, and Soul Size.

Staves are effected by Magic Skill.