Your tween probably wants to feel more grown up by staying up later, but sleep deprivation will damage her health and school performance. Tweens live in the moment, and aren't fully able to appreciate what the consequences of staying up too late will be, such as falling asleep in class the next day.
Letting your tween stay up a little later on weekends and during summer break won't do any harm, but keep in mind that, while adults need about eight hours of sleep and teenagers need about nine hours, the average tween needs closer to 10. Sleep deprivation can lead to mood swings, trouble getting up in the morning and even learning and attention problems. And we know you have enough on your plate already raising a tween, without all that.
If your child is really eager to try a later bedtime, push his bedtime back half an hour on a trial basis, with the understanding that you will change it back if he shows signs of too little sleep. He may think you're being "lame," but this is about his health and well-being. (Don't expect him to understand that though.)
Should limits extend to your tween's friends? We'll weigh this one next.