I'm in a mixed marriage. My parents, the same people who raised me to not be racist, voiced their opposition when I told them about our engagement. Until then, when we were "just friends", they had no issue. I told them I was disappointed in them and that they were, in fact, racist. Furthermore, I told them if they ever said a disparaging word about our marriage again, they would be cut off. They silenced themselves and have behaved as I had initially expected ever since. It's been 8 years.
I'm sorry, but your friend is racist. She may not have considered herself racist (like my parents), but this is probably the first time she has been truly tested. She failed. The good news is that it's not terminal. This is actually an opportunity to look inside herself and examine what exactly makes her uncomfortable. I never asked, but I'm sure my parents had to do a lot of self-reflection. While they had black friends, they had never had anyone come that close before.
Yes, mixed couples can have difficulty, but it's a lot better than it was. I know many mixed couples and some of us have never had any real issues while others have had some. It's a risk you take and life is filled with them - no guarantees.
Children actually do not have trouble with mixing; it's the parents who do. I've never met a racist child; it's something that is taught. Your friend should be proud that she raised a daughter who is not racist. Now she can learn from her daughter. And, honestly, these are teenagers. What are the odds that this relationship will last a lifetime? As someone mentioned, the mere fact that she voiced such strong opposition will only reinforce her daughter's desire to be with this boy.