The photo makes it a little awkward to see but if that is a a ZIF connector (zero insertion force) ie; the cable is pushed in and you slide a couple of locking pegs across then the fix isn't easy.
When you read apple's repair manuals and they mention these ZIF connectors any damage to them usually esults in the entire socket needing replacement (normally thy just replace the motherboard). What happens is the locking pins break so the cable isn't held in properly. If this is what has happened to you then you have a couple of choices;
1. Finding a replacement connector is one thing fitting it is quite another and unless you are seriously talented with a soldering iron you'll be looking for professional help which will cost.
2. A number of repair centres do logic board exchange programs, while this may see like an expensive step the fact they could repair yours means you will likely get a good trade in price for it. On the plus side they could fit a later Motherboard so you may even be able to get a slight speed increase for your money.
3. Now this option may seem like a bodge but if it really is a case of the connector simply not staying in the socket you could repair it yourself. Not by soldering but by wedging the connector in place, testing and the if necessary using glue to fix it in place. Have a play around see if you can wedge it, see if the display will work and then decide if you want to fix it permanently. If you use a good quality super glue carefully it can always be removed later should you want a professional fix (the decision needs to be yours though
You need to way up the pros and cons of each option. You will probably find that parts and labour are more exepensive then the machine is currently worth. Doing it yourself if you find the part is an option but if something goes wrong you've just lost all trade in value and the machine is completely unusable. I may have the original Ti DVI apple repair manual kicking around on my hard drive, I'll have a look later if that would be of use to you.
