After we broke up, she cried about me to her friends.
After we broke up, she cried over me to her friends.
Is using about grammatically incorrect here? Is there a difference in meaning between these sentences?
They are both correct. However there is a niche difference you ought to know.
If you cry about me to your friends, typically, you are talking about our relationship while crying.
If you cry over me to your friends, typically, you are just sobbing.
Even though there are interchangeable, you might want to use about in some situations and over in others.
Both are grammatical. I think cry over is especially used when someone is sad about loss: crying over a former romantic partner after a breakup is a good example. A specific case is the idiom (don't) cry over spilled milk (meaning, don't worry about losing something insignificant).
Crying about something or someone simply means the reason for crying (and the feelings behind it) involves them.
In some contexts over and about may both be appropriate. But over is especially appropriate in the context of a breakup, death, or other painful loss.
Removing any ambiguity:
After we broke up, she cried to her friends about me.
After we broke up, she cried to her friends over me.
They both mean the same thing.
To cry about something
To cry over something