Would a qualifier such as “Any naked-eye star named before the invention of the telescope” be helpful in your query?
ISTM that every star with an inhabited planet orbiting it has a name. And the name, in the local language, is: The Sun.
How about this as a criterion: No names which contain a number or the equivalent of a number. That would rule out things like Wolf 359 and epsilon Eridani (since the epsilon there really just means “fifth”), but would let through “Armpit” and “Foot” no problem.
That’s mostly because they’re borrowed from Arabic and start with the Arabic definite article “al-”.
Well, that’s a point. But if Alpha Centauri is out, then names which are just foreign words for the equivalent of Alpha Centauri are also questionable.
The OP asks a rather unanswerable question, then. But the answer is somewhere between 60-300, given the rather vague parameters (not his fault, it’s a rather… cosmic question.)
Note, dudes, that Cal is a well-known expert on mythology. 
I found these lists intriguing: had you ever heard of Sadalsuud, Miaplacidus, or Bellatrix?
There’s several stars named after some astronomer who thought the star was interesting for some reason:
Barnard’s Star highest proper motion star
Kapteyn’s Star second highest PM after Barnard’s, but highest when discovered
van Maanen’s Star white dwarf with high proper motion
Luyten’s Star not sure why this was named
Teegarden’s Star initially thought to be very close to the sun (about 7.5 ly away), but turned out to be further (12 ly). This was named in 2003, so is probably the most recently named star.
None of the above are visible with the naked eye, but then there’re Sualocin (alpha Delphini) and Rotanev (beta Delphini). These names are something of a practical joke. An astronomer took the Latinized form of his first name and reversed it to get Sualocin and did the same with his last name for Rotanev.
Along the same lines, either gamma Cassiopeiae or epsilon Cassiopeiae (Wikipedia says both were, so one of the pages is wrong) was named Navi by astronaut Gus Grissom by reversing his middle name. The name is kept as a memorial to him (if you remember, he was one of the three astronaut who died in the Apollo 1 fire). The other two astronauts also have stars named after them: iota Ursae Majoris is named Dnoces (second backwards) after Edward White, II and gamma Velorum is named Regor after Roger Chaffee. I believe these three stars were all navigation guide stars used in the Apollo program.
I’d heard of Bellatrix, but only because it was a random name that popped into my head when I was writing a novel (which I’m not sure if I’m still working on or not.) So I was wondering where I got the name from, and sure enough, there’s the star. Plus, Bellatrix is a heavenly woman warrior, so the name fits that way as well.
Then of course a year or so after that, a certain Harry Potter book came out. Thankfully, my Bellatrix is nothing like hers other than being combatative.
And here I’d thought that everyone had heard of Bellatrix. All of Sirius Black’s family are named after stars.
Well, there’s Lana Turner, Ava Gardner, etc.
There’s a star called Narcissa?
I think a combination of these two sets of criteria would pretty much pin down an actual list of stars that have at least one name. Sapo, would this pretty much describe what you’re looking for?
Well I didn’t realize my question was so complicated when I asked but yes, that’s pretty much what my OP has evolved to.
Ancient Arabic for “brightest eye of the fox” is fine. “Eleventh star of the turtle” is not, even if said in Latin to make it sound ancient.
I am Ok with stars named in modern times and not visible without a telescope, though. If a star is unique enough to get a name when the other 597,394 stars cataloged that year only got a serial, that’s gotta count for something. So Barnard’s and Pistol are in.
Okay, I could get a translation of “brightest eye of the fox,” but vBulletin does not display Arabic script. The other would be Kappa Turturis, or perhaps Kappa Testudinis.
WARNING: Here be Zombies
Another star has recent;y joined the above list:
Scholtz’s Star Another one named for its unusual proper motion, but this time because it’s very low, rather than very high… Low proper motion means it’s headed either directly towards or away from the Sun. In this case, away. In fact, some 70,000 years ago, it passed less than a lightyear from the Sun.
This is a very dim red dwarf plus a brown dwarf. So dim that even at its closest approach, it was not visible to the naked eye. It’s now about 20 lightyears away.
Someone has fixed up epsilon Cassiopeiae’s Wikipage to say that it’s not the one named Navi, even though some sources say it is. I do wonder if that was due to my post.
Names of places have all been made up by someone. Does ‘New York’ not count ads a city name? What about London (originally an insignificant town called Londinium)?
Lots of villages in the UK have prefixes like ‘Upper’, ‘Lower’, ‘Nether’ and so on - it doesn’t mean that Upper and Lower Slaughter are not separate places.
Not even close.
If you can lay your hands on Star Names: Their Lore and Meaning you can decide for yourself how many stars have ancient names in various cultures. Fascinating if not entirely up-to-date as far as the accepted names go. (Accepted by whom is another good question! 
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Rocky Mozell has them all in book form in the U.S. Copyright Office…