What is the difference between those sentences?
- I'm on your side.
- I'm at your side.
- I'm by your side.
The meaning should be the same: I'm with you. I will support you. I will not leave you hanging. I will help you etc...
"I'm on your side" only indicates that I am in favor of your cause, that I am not working against you.
"I'm by/at your side" carries the message you want, that you are right there to provide whatever support is necessary.
I prefer "by."