Sorry, but I have to chime in here. As almost EVERYONE has said DO NOT LOAN/GIVE THE MONEY!!!!!!! Also as one of the previous posters has said (sorry forgot who it was) you will LOSE your money because this is to prove that they can pay for school, and they obviously DON'T have money for school.... so I wonder where this money is going to go...
It is NOT your responsibility or duty to help this "cousin". It is extremely RUDE to even have asked you in the first place because loans of this size should only take place between DIRECT family members not cousins of your gf. I am asian so I do understand this trust issue, but why do you have to follow it? 10k is too much to trust to someone you don't even know. The only benefit I could see from this is that you would be maintaining/strengthening the relationship with the family on a whole and especially so with the mother. BUT this is only initially, most likely your relationship with the family as a whole will become very strained.
I say this because you must remember that generally money RUINS relationships not help them, that's why people here are saying if you are going to give/loan money don't expect it back. The reason why I say that it ruins relationships is that if you were to lend them the money and they don't pay you back, which most likely will happen since they can NOT do so since they have to spend it, you would most likely feel cheated/disconnected from the family and you would not only feel anger towards the family, but they would also feel anger towards you since you should not be asking for the money back. So basically you lose all the ground you made by originally lending them the money, but also you lose $10K!!!
Now lets say you do lend them the money, well guess what... you are now offically the bank of the family and you WILL be hit up for more money. The problem will be that you really can't say no without causing a LOT of anger, animosity, and jealousy. This is because the other family members will wonder why you only lent the cousin money and will not lend them money also. Believe me you do not want to become the bank of the family it will suck so bad and you will most likely break out of the relationship thus making the originally loan which was to better the relationships a useless and very costly gesture.
Also if your gf breaks up with you(which you said would not happen, but lets pretend) because you didn't lend the money then she's obviously in the relationship for the wrong reasons and it was better for both of you. The mother would most likely be angry at you, but I think it is more important to do what is better for YOU and not the mother because it is YOUR money.
This is a very tuff decision because I am pretty sure you are getting a LOT of pressure to lend the money, but just because everyone says to jump off the bridge does not mean you should

So basically if I were to put my whole argument in one sentence it would be: DO NOT FRIGGIN LEND THEM THIS MONEY BECAUSE IT IS NOT WORTH IT AND WILL CAUSE YOU MORE PROBLEMS THEN IT IS WORTH AND IF YOU DO LEND THEM THE MONEY YOU ARE 100% WHIPPED AND NEED TO GROW SOME BALLS!!!!!!!!!
Good Luck!