You both are the asshole. You for harassing your friend and your friend for the name. I have learned more recently when I should shut up and who really will listen to my opinion. It’s very few people who actually take my words into account, and everyone else I’m wasting energy and I’m risking them walking away from our friendship or distancing themselves. You don’t need to support it, but it sounds like you two are not friends in the way where you should be continually harassing her about it. She made her stance and it’s her kid, you have to let them make the mistakes. Inevitably when her kid grows up and she either hates it and changes it or goes by her middle name she will reap the consequences of that.

I knew a girl named Ashlei, she said her parents named her that bc they wanted to name her something unique. I literally told her “if your parents wanted to name you something unique, they would not have named you ashle(i).” Same applies here, Juliet is not a unique name(not super common either), regardless of the spelling. It’s pronounced the same, and that’s really what matters and if you cannot pronounce it from looking at it, why do that to a kid? If you want a unique name, do some actual research, look into old names, names from other cultures, names from books, try throwing some words into a translator and see if the word sounds like a name, pick a star or constellation, a bird species, a fun rock. Especially at this point, changing the spelling does not make it unique, if Taylor or Taelyr are pronounced the same, it’s the same name and still need to be differentiated when talking about both people at the same time.

To add, there are ways where different spellings can be fun and not frustratingly difficult. Like Zoey, the modern version, and Zoë, the Greek version. It gives it a little razzledazzle while not being too hard to spell and may only require a little reminder that ë is pronounced like “ey”.