My mother wears hijab but not properly. It often falls off and she doesn’t seem to care. It hurts me because I’m watching my own mother sin in front of my eyes. Is there anything I can do islamically speaking to get her to wear it properly? She is my mother so she has authority over me and if I say anything she may get offended by me.

You must maintain respect and dignity towards her. You may be able to make the point indirectly to her (by recounting a story of Sayyida Fatima or Zaynab) or do so in a gentle manner.