Hi, I'm Dack from Messores Republicae and this is what we do for our raiders. We are a highly selective 16 man progression raiding guild.
When we bring people in we tell them that they'll get their chances to show us how good they are and what they can do as far as mechanics go. DPS really isn't a concern when seeing how well people can do because we have people who have a high expertise at their class and are more than willing to help out anyone that needs some coaching.
So on to the important stuff, mechanics! When we're trying out a new member we take he/she into an operation of their gear level and asses how well they can do what they need to do. DPS not being hit by red circles, using threat drops at the right time, you know, the things people need to know and be able to execute in order to be competitive. If they can't get it down in a reasonable amount of time then they're rotated out of the operation for someone else to come in so that the other 15 people aren't having their time (which we're all paying for, mind you) wasted. As soon as we run the raid again and get to that mechanic though we'd pull he/she back in to run it again.
So, let's say we're in your situation and we've got this awesome guy that's come along and been a part of our guild for a long time; it just so happens that he just flat out SUCKS at mechanics. Well, we'll run him and run him and run him week after week on the mechanics he can't get down until he either learns them or gives up. If that person really wants to be here and improve so that they can be part of the team and pull their weight then they'll keep trying and trying until they learn what to do and develop raiding skills. Practice makes perfect, and even though some of us learn faster than others none of us were born with the ability to run raid mechanics. I remember back when Colonel Daksh used to one shot the tank if he were hit by the boss's laser eyes. Our tank just couldn't get it down so my other guildmates and I would sit on our speeders and just watch him do the dance for hours each day until he became proficient at knowing when to move and how to move his character effectively.
Gist of it:
1) Have them find out where they're deficient
2) If it's mechanics have them run the mechanic they suck at over and over again but rotate them out of the operation after a good number of tries so that the other raiders don't feel like they're having their time wasted/getting uber bored
3) Have people help train this person with a spec/rotation/stat priorities
If you're person is serious about being with you guys and pulling his/her weight then they'll log in and work on improving themselves. Remember, you're not doing this to be a dick, but to push people to be the best they can be as well as making it fun for everyone!
Just what we do, if any of this helps you decide what you should do then that's awesome, if not, then that's cool too