Well if nothing else, here are two name generators specifically for FFXIV you can try. They offer more diverse output than the in game name gen, but are still lore friendly.

https://www.fantasynamegenerators.co...antasy-xiv.php
https://ffxiv.pf-n.co/name-generator

But... even if it's a struggle to think up a last name initial, you might find in the long run it helps. Names can be much longer these days, usually, and often spaces are allowed - and when not, short names are often taken. Having a last name to throw in, even if it were something like "TerinSmith", can help with all those issues should you try other online games here and there.

Personally, sometimes I just pick a name (as in just the last name, perhaps) from a recent book I've read - especially secondary characters. Some of them sound plausible without being too exotic which I like. Especially the Hyur namegens above could also potentially help with that. Similarly, I've done what Vahlnir does sometimes... make the name a very obscure reference to something I enjoy, possibly permutated beyond any reasonable recognition.

And on that note, here's a different sort of name generator: https://www.spinxo.com/
This one generates a name using readable words based on... well, whatever you tell it you like, and it also has a built in thesaurus so it'll automatically pull in similar words to its pool as well.

And, that's about as much help as I can offer. It's a bit of a personal thing - I think you'd agree, from the tone of your post? And we just don't know you, let alone for the years it might take to know you well enough to make a good direct suggestion. I hope this is enough to help guide you to your ideal name, though. Good luck!