Why did he mom file a paternity case against a man she never met?

It was the daughter who put her up to it.

The writers are dumb and they tried, in both movies, to incorporate things from the books and make them even bigger, while missing the simple things like - how the hell did she even know him.

In the book, the paternity suit as bogus, made up by the people after Turner. Sam and her mom knew nothing about it.

Fighting a religious war is like fighting over whose imaginary friend is better.

I thought this movie was really stupid, but now I feel dumb for not realizing how that made no sense whatsoever.

Didn't they say something near the beginning that the mom used to be a hooker and she's just getting clean from drugs?

Maybe she spent a good deal of her prostitution time hanging around military bars or something. And she and Reacher did meet in passing at some point. Like maybe he was at the bar with a group of guys and even though they didn't sleep together, she just happened to remember his name somehow.

So when Samantha encouraged her to file the paternity suit against her father. She didn't want to tell her daughter that she didn't know exactly who her father was because of the drugs and multiple sexual partners, so she just named the only guy's name she could remember from back then.

Reacher wouldn't necessarily remember the face of every hooker he met especially if they didn't sleep together, and she probably looks a lot more clean cut now, and if she was on drugs she might only remember his name not his face. Hence them not recognizing each other.

And just like that I officially put more thought into the plot line than the writers did probably.