I've enchanted a couple of items and thought I would give it another try. This time, I chose an iron dagger I had crafted and refined, a common soul gem and soul trap. Everything was bolded so I figured I was good to go. I thought I would name this one (hadn't tried that before).

No matter what I do, it says this item can't be enchanted. I switch daggers, I switch soul gems, I switch from soul trap to absorb health. What am I doing wrong?

Did you hit enter to fix the enchantment before selecting the weapon?

After renaming it you have to hit and after selecting the enchantment - in your case soul trap - you must hit , too.
It's not really intuitive, because when you choose another enchantment, which is no offensive power, you must NOT hit to confirm the enchantment.

Naming enchantments is a bit pesky, but once you get the hang of it, it helps you to find the ones you make in your inventory/containers.

after you finish renaming the weapon you must then CRAFT the weapon. on the 360 hit the X button.