If the other friend is close to your daughter in terms of interests and academic abiliity, I definitely recommend looking at Story of the World or even something like Five in a Row. It's not overly academic for a bright child, and the craft projects might be perfect for this type of playdate. A lot of girls also connect with the related cooking projects.
Otherwise, is the friend's mom around, too? If so, I would consider asking her for activity input. If she isn't, I think you might be on the right track staying hands-off and letting the kids come up with their own activities together. At 6, they're capable and will likely have a great exercise in creativity along the way. My DS6 has been putting together a "wizard's camp" with his friends, and I've seen a lot of quasi-academic ideas put together. If they choose their "subjects" together, maybe that will help?
Otherwise, I'd continue to monitor the friend's feelings. Last year, I had then-DS5 and his two 5 year old friends over for a day and tried to homeschool together. They had all started K in PS together, then all started HSing in Jan. The twins' mom is also an ex-elementary school teacher, so I guess I made some erroneous assumptions about what they could handle. I tried to REALLY simplify what we were doing. Since DS and I were studying Africa in Story of the World (SOTW), I read them an Anasi picturebook (keep in mind DS was reading Harry Potter on his own), with a SOTW map exercise (extremely simple for DS), and tried to implement a cooking and craft activity. The other kids didn't have the attention span to make it through the book, couldn't grasp the idea of Africa as a continent, had no interest in cooking, and were itching to run outside and play before we made it to the craft. Even though DS was very interested in helping them, I realized that this would be defeating to their self-esteem if we tried to continue.
DS also tries to play math games and help his friends with math problems. He's darlingly sweet about it, but again, I try to lead them to something non-academic. I want him to connect with his age-peers, and I don't think sharing academic interests when he's so far above their level is not going to foster those relationships. If a friend of his wanted to....say play football but always won no matter what, I'm absolutely sure that my son wouldn't want to play it over and over again.
