Some of the ideas here are good, but you're honestly probably not going to convince your mother to accept you via debate. -_- I didn't "act female" as a child, but I wasn't exactly given free reign to do that. Plus there's not one way female and male people act, we can ALL find "signs" in our childhood if we look hard enough but realistically we could find those things in other people's childhoods too. Someone isn't trans because she liked barbie dolls, someone doesn't stop being trans because she likes video games. I mean if liking video games makes you a man then half the people who play video games must be ftm

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I'm not sure your age, but I explained to my parents that:
-Them not being aware of something doesn't mean it didn't exist or didn't happen, but...
-There's not necessarily any reason they SHOULD have been aware of it, given that I was trying really hard to hide it.
-It's not something they "caused" or could have prevented.
-I had a lot of reasons to hide it in regards to social pressure, already dealing with harassment and not wanting to draw attention to myself, etc.
-That at this point it wasn't a question of me transitioning or me not transitioning, but me transitioning with their support or transitioning without it.
That basically did it. I think it would have been harder if I was still a minor, but since I'm an adult basically they got to choose between being one of the parents who estrange their children because the child is LGBT or gradually changing some parts of how they relate to me or view me. I'm still not sure if they're totally comfortable with it but they ultimately went with the second one.