Is it normal that my mom won't let me spend the night w/ my boyfriend?

My best guess she see's you staying over there as announcing to the world that the two of you are sleeping together.
If you two just fuck someplace and leave, people might just think you two were just hanging out but if you stay the night, anyone who see's you two leave would automatically assume you are having sex.

Now if that sounds crazy, I agree it kind of is but look at it from your mom's point of view. I don't know your mother but from the way you've put things I'm assuming she doesn't like the idea of you two having sex but she understands you're going to do it. What is probably upsetting her so much is that she is afraid If you stay over it will become undeniable to anyone that you are sexually active and she's afraid of that information getting out and people thinking that you are a slut and if they think you are slut they will disrespect you.