There’s two sides to the “no one owns a name” scenario. It’s one thing if it’s a random co-worker from a job you had 10 years ago (in which case it’s more than likely not copied, or at least it doesn’t matter anyway because you won’t see them), but it’s another when it’s someone you’re going to spend a lot of time with.
I was the first one in my group of friends to have a child, but I hadn’t been, and my two DDs’ names had already been picked, I would have picked something else. I would also be annoyed if friends chose the same names as I did as there are literally thousands of names, so I find it hard to believe someone couldn’t find another single name they equally liked (obviously if it was a family name it would be different, perhaps still annoying but you’d understand and get over it).
OP, I think you’ve been more than generous over the years, but unfortunately I think you do need to put your foot down this time. She will still likely copy the child’s name and everything else you do, but at least if you take a step away the child being called x won’t matter to you as much on the day to day.
I’d probably reply to her message with something along the lines of
“Hi x. I’ve had a bit of a think about what you’ve asked me and I’ve came to the conclusion that it’s better to just be upfront and honest. I would absolutely mind if you used x’s name. It’s a name that my partner and I chose together which has significant meaning to us, and if I remember correctly, you didn’t even like the name when we told you. It would be odd to use it considering the closeness of our friendship and I’m sure there are many other names which you would like more”
If she tried to convince you she had always liked the name etc then I’d respond with something harder hitting, ie
“Okay, I appreciate that you now want to use the name but unfortunately if you do then I will have to take a step back from the friendship. This isn’t the first time you have taken something of mine and copied it pretty much exactly; the engagement ring, etc etc etc. I haven’t said anything before as honestly, those things didn’t matter as much to me and I was giving you the benefit of the doubt, but the name is a step too far in terms of what I’m willing to accept. At the end of the day, the decision is yours to make but you know exactly where I stand on this, and I won’t be swayed or talked down”.
If she insists on the name, I’d walk away. If she wises up, I’d keep her at arms length until I feel she has changed. If she doesn’t change, then the friendship is dead in the water.
Taking inspiration from a friend is one thing, but copying every single thing they do is weird and regardless of any MH condition, it’s not a nice experience for you to have to contend with. You’ve been very kind but enough is enough.