Try asking your mum why she doesn't want you to stay over. Maybe if you can explain to her that whatever it is she's worried about won't happen, she might feel better about you going there. It might also be a good idea to have both of your families meet up for a day. Then she can see that they are nice people and might let you stay over there.
If that doesn't work, maybe suggest that your friend stays with you for a night. That will show her that you are really good friends, and that your friend is a nice person. I hope it goes well for you :)