I personally feel once you tell one person, you tell everyone, so I never tell anything I don't want other people to know. If I had a relative that I knew couldn't keep a secret I would never tell her a secret, even if it was my mom. So for telling your mom, I do think you need to understand that you cannot control her and if you know she is a blabber you should have expected it. Of course, your mom is very rude and now you know, even if you beg, your mom still cares about herself more than others. Use that info for the future.
I would rather be told something last than keep a secret just because sometimes things randomly come up in conversation and secrets slip out. I don't want the responsibility for other people's secrets. If you want something to be a secret, just keep it a secret from the beginning.