13th Birthday Ideas?

My daughter is turning 13 this August and she has been looking forward to this birthday for a very long time so she wants it to be perfect. She is not girly at all - she's very into superheroes and highly dislikes any kind of beauty activities.

The main problem is it has been a rough year for her. She doesn't really have any friends that she feels close to. She is always complaining about how critical and brutally honest they are. She is very lazy and doesn't do any kind of sports. So, because she isn't a part of any team or club and it is summer, there is not a very good chance she'll make some friends by her birthday. I tried convincing her to just invite the 4 or 5 girls she knows, but she's worried about what to do. Obviously she wants to have fun and do activities she enjoys, but these other girls don't share any of the same interests as her. Everyone last year (which was the same girls she knows now) at her birthday party was just kind of bored the whole time. I'm not a big social mom so I don't really interact with the kids. My daughter is also very stubborn so it's near impossible to plan any kind of organized party/sleepover.

We have considered just a party with family, but we are not all too close with most of our family and the few cousins she is fairly close to are much older than her and would not really want to come. Please help. She is freaking out about this constantly which is definitely putting stress on me to fix this. Our family has a short budget and a fairly tiny home. Along with my daughter I also have 3 sons, all 18+ who live at home and will get in the way of her party. She would prefer a sleepover even if it was only with one of the people from the party. Her birthday is August 3rd (we won't do the party that day - that is for celebrating at home) and school starts back up August 11th so we also need a good date where people won't be on vacation. Any help or suggestions would be greatly appreciated!