I should start by saying that this question is inspired by a Calvin and Hobbes cartoon that was e-mailed to me today.
Would it be possible to transplant a third eye into a human being, hook attach it to blood vessels and a third optic nerve, and have it work?
Despite the ugliness factor, I would love to have an eye at the back of my head 
My mom has one of those.
If it worked why would blind people choose to stay blind?
The same reason some deaf people chose to remain deaf: Because they were born that way, or they like it.
It sounds crazy, but there are folks out there who are deaf, who could have hearing thanks to a cochlear implant, but chose not to. There’s even been a case (mentioned heavily on these boards) where a deaf lesbian couple chose a deaf father for their child in hopes that the child would be born deaf. There are deaf activists who are totally against any efforts at enabling the deaf to hear, even those who weren’t born deaf, but became deaf due to accident or disease. They feel that there’s a deaf culture which would die out if there were ways to correct hearing loss.
To answer the OP: No it would not work. For a tremendous variety of reasons. The foremost one being we do not have anything near the technology allowing us to cause the brain to grow a new optic connection that we could jack into.
QtM, MD
Even if we could somehow add another optic nerve, the brain probably wouldn’t know how to combine the input with the other eye’s images.
Just because you plug something into your PC, that doesn’t mean there’s software able to use it. And I doubt the brain is “Plug And Play”…
There IS a Deaf culture. It’s a huge, huge, big deal with lots of documentaries about it. Deaf parents who are part of the culture want their Deaf children to stay part of the culture. They want other people’s Deaf children to be part of the culture, too. Trust me, this is huge.
As for the OP I would think that it would be the brain that wouldn’t accept it. Look at how many operations go wrong because the body/brain won’t accept the new “stuff;” why would the brain be flexible enough to accept a new eye? It would probably reject it.
Clearly adding a third eye wouldn’t work, but what about moving one? Would it be possible to extend or stretch the optic nerve and move your left eye to the back of your head?
See above. the part of your brain that combines your eye’s separate images into one would most likely get very, very confused. Getting around this would likely require the ability to grow new brain structures, which as mentioned we’re lightyears away from being able to do.
With practice I think you could sort it out. It would really be no different than the image you see when you put your hand between your eyes and place a photo or some picture such that only one eye can see it.
You’d lose all depth perception, but you could probably adjust to the image. The biggest hurdle would be to train your eyes to focus independently of one another. Anyone know if this is possible?
JFTR, the pineal gland is light-sensitive, and in some creatures (the tuatara Sphenodon punctatus comes to mind) it is located just below the (translucent) skin and can perceive the presence or absence of sunlight, etc. – useful for basking purposes. Some have referred to it as a “third eye” in consequence of this, though that’s a massive exaggeration.
If not eyes, then what about limbs?
I’d enjoy having a couple extra arms.
I don’t see why the same logic wouldn’t apply to limbs. The brain has evolved to handle 2 arms, and slapping another one on without changing the brain the handle it simply wouldn’t work.
The OP question asked ‘is it possible?’,
and the answer to that is double.
Is it possible now or in the near future?
Not by a long chalk.
Will it ever be possible?
Barring accidents, yes.
If medical science, neurological investigations and nanotechnology develop as expected, you could have a hundred eyes, each watching a different TV channel if desired.
Sc-ifi worldbuilding at
http://www.orionsarm.com/main.html
Eh, guess not 
Never said it wasn’t. I’ve got a cousin who is so profoundly hard of hearing that she might as well be deaf. She, however, has done what she can to remain mixed into hearing society and doesn’t believe that if it’s possible to gain hearing, one should stay deaf.
I asked a question similar to this one a while back; here’s the link.