I'm going to ask my mom to drop my friend off when she was picking me up
Why would you ask your mom to drop your friend off?
My friend's mom does the same for me so I want mine to do the same for them.
It's good of you to try and pay people back for the things they've done for you. Your mother may develop precious bonds too as a result of these favors.
I hope she does but it isn't guaranteed.
I think you should talk to her about it first. She could have other plans she needs to do so don't be rude and ask before.