Try, try again. As hard as it is to face rejection, sometimes the person you are hoping to befriend really did have plans the night you asked her to join you for dinner. Ask again. In the end, the relationship you might create will be worth much more than the embarrassment of a repeat rejection.
You may have noticed that making new friends is often scary – you expose yourself to rejection when you try to befriend others. On the flip side, the rewards are plentiful: a longer, happier, and less stressful life. In the long run, I think it’s safe to say that it’s worth it. Therefore, consider making your extremely busy life just a little bit busier and making a little bit more time for friends.