Not sure how uncommon you're looking for, but I think I already did this.
I chose Sage.
According to some quick googling (read: not sure how accurate this is), Sage wasn't in the top 500 names for my birth year, but my given name was top 100.
Though both names I've used have been trending towards more common for most of my life. Looks like my original name is now a top 35 most common, but Sage is still outside the top 100 for females and outside top 300 for males.(Really wish there was more data for the popularity of nonbinary names. I think Sage is probably one of the more popular enby names, so maybe it's not all that uncommon, depending on how you define it.)
I've already got a name that puts me quite comfortably in the 1% of names used in my country. It's not as cool to have a uncommon name as you think. It's easy to get picked on, you never can buy those souvenir mugs or keyrings with 'your name' and it gets misspelled.
I mean I've written a book where my name is on the cover and one review managed to misspell it...
Otoh. I once met someone on holiday in Ireland that said 'i one met a kid with that name and it was in Croatia, about ten years ago'.
That kid was me. If I had a more common name we'd never have known.
I already did, but I won't be sharing it in a public forum :P
Doesn't apply for me, but nonbinary folk are great for this question because they all name themselves after random inanimate objects (which is 100% valid)
I’d probably pick the unique name of someone very famous, in the hopes that I could still hide myself online by having references to me shoved 10 pages deep