What is the farthest-away star visible to the naked eye? My girlfriend and I were watching Cosmos, and something Carl Sagan said got us wondering what the farthest-away visible star is. Obviously "visible to the naked eye" is a fuzzy concept that might have many defensible answers, but hopefully not too many.
To make the question a little more interesting, let's restrict to individually distinguishable stars; otherwise the answer is pretty clearly some Local Group galaxy, and there aren't many of them to check.
The closest thing to a reasonable answer we came up with was this wikipedia list of stars that are more luminous than any closer star. The farthest-away star on that list with a plausibly visible apparent magnitude is Eta Carinae (7500 ly away, magnitude 4.55). However, there are several reasons why I'm not willing to consider this definitively answered:


*

*It's a wikipedia article, and a poorly sourced one at that. So I don't entirely trust it.

*It sorts stars by bolometric luminosity rather than visual luminosity, so perhaps there's some farther-away star whose spectrum is better-centered in the visible range.

*The farthest-away visible star isn't actually guaranteed to be on a list of that sort, even assuming the other two points are cleared up. Perhaps the farthest-away visible star is only barely visible, and there's some star both closer and absolutely brighter than it which makes the list.


Given all these points, is it actually the case that Eta Carinae is the farthest-away visible star, or is there some visible star that's farther from us?
 A: Following the questions raised by Rob Jeffries, I have completely rewritten my answer:
What is the farthest-away star visible to the naked eye?
Indeed, this question has many defensible answers. It is not just the concept "visible to the naked eye" that is fuzzy. The stars we see are seldom single objects, but rather binary and multiple star systems. Do we allow doubles and multiples? The wording of your question ("let's restrict to individually distinguishable stars") suggests to exclude multiple star systems. But the candidate farthest visible star put forward (Eta Carinae) is exactly that: a multiple. And like most highly luminous stars Eta Carinae is also a variable star. Should variable stars be allowed? Should we allow variable stars that in the recent past were visible to the human aye, but currently are not? If so, do we also allow cataclysmic variable stars? Do we allow novae and supernovae? Apart from all these ambiguities, as stressed by Rob Jeffries in below comments, there is also the issue of (often considerable) uncertainty in cosmic distances. How do we handle these uncertainties?
Let's first define what we mean by "visible to the naked eye". Which stars are visible to your naked eye depends on the light pollution of the site you are observing from and the atmospheric conditions (and obviously also on you eyesight). A so-called "magnitude 6 sky" is often taken as the standard for a good dark site with no light pollution. The threshold stars you can see in such a night sky have apparent magnitude 6. 
So we can eliminate a key ambiguity by changing the question into "which star brighter than 6th magnitude is farthest away?". 
According to this article :

"The farthest star we can see with our naked eye is V762 Cas in
  Cassiopeia at 16,308 light-years away. Its brightness is magnitude 5.8
  or just above the 6th magnitude limit."

This answer puts forward a variable star, but clearly excludes supernovae as that would have resulted in much larger distances (more about that later). Rob questions the apparent five-digit accuracy in this answer. A bit of research reveals that the distance figure is derived from the central value in the measured parallax of 0.22 +/- 0.59 mas (milli-arcseconds). This means that we have no more than a 50% confidence that the distance is indeed 16 kly (kilo lightyear) or more. 
We should not blindly accept a 50% confidence level. Rather, we should agree on a confidence level that is deemed sufficiently strict for the intended purpose of selecting the most distant star. Yet another ambiguity to resolve! 
I propose to use the one standard deviation upper range of the parallax measurement (in the case of V762 Cas 0.22 + 0.59 = 0.81 mas) to derive distances. This gives us an estimated distance of 4.0 kly with a confidence of about 85% that the actual distance is at least this value. (As Rob points out, a more recent parallax measurement for V762 Cas results in 1.18 +/- 0.45 mas. If we would combine both parallax measurements to derive a chi-square estimate of the actual distance, we arrive at a value compatible with 4 kly.)
This results in the conclusion that the often quoted V762 Cas (see e.g. here and here) is unlikely the most distant naked-eye-visible star. For instance, HIP 107418, put forward by Rob as candidate most distance star, has a lower one standard deviation upper range of parallax of 0.62 mas, corresponding to a 85% confidence distance of 5.3 kly.
I do not have the means to analyze extensive star data bases, but offer this candidate most distant naked-eye-visible star: AH Sco, with a one standard deviation upper range of parallax of 0.48 mas, leading to a 85% confidence that its distance exceeds 6.8 kly.
Finally, what answer do we arrive at if we allow for a broader range of variable stars, including supernovae? 
I propose SN 1885A at a distance of 2.6 million light years (!) as the most distant single star that was once (almost 130 years ago) visible to the naked eye.
A: Rho Cassiopeiae is my favourite candidate.  I know it is a variable but is visible at present on very clear nights. One of the reasons for choosing it is that it is listed as being 8,200 light years away (further in some lists) which means that light was on its way from it 6,000 years ago when some say the universe was created.  This would require the light to have been created in transit. Cassiopeiae is easy to spot and Rho is slightly up from the top of the W when the Big Dipper is below Polaris.  
A: I have taken the revised Hipparcos parallax catalogue, produced by van Leeuwen (2007, Astronomy & Astrophysics, 474, 653) and taken a subset of stars with Hpmag <6 (i.e. roughly the naked eye limit) and accepted only those objects with a parallax/error in parallax > 2.5. Anything with a larger fractional error in parallax really can't be trusted.
If I then look at this list, sorted by the reciprocal of parallax I see a number of candidates for the most distant naked eye star. I will list the top 4.
Name          Hpmag  |  Plx (mas)  |  e_Plx (mas) | Distance (pc) | Name/Spectral Type

HIP 107418    4.39      0.48           0.14         2080            nu Cep  A2 I  
HIP 22783     4.29      0.52           0.19         1920            alpha Cam  09.5 I  
HIP 54463     4.09      0.52           0.17         1920            chi Car  G0 I   
HIP 107259    3.91      0.55           0.20         1820            mu Cep  M2 I  

Note that V762 Cas (=HIP 5926) has a distance of only 850 pc (parallax 1.18 +/-0.45 mas) according to this catalogue. The oft-quoted Deneb has a parallax of 2.31 +/- 0.32 mas and thus is likely closer than 1000pc.
The top 4 are all blue supergiants or yellow/red hypergiants. The last one on the shortlist is a well known and very well studied object. The size of the error bars is such that it is hard to say which (exactly) is the most distant, and there are another few further down the list that could be more distant within their parallax uncertainties. The Gaia results in early 2017 will resolve this issue.
Eta Carina does not have a parallax in the Hipparcos catalogue, but this maybe because it was too faint at the time or more likely it is not a sufficiently point source (surrounded by nebulosity) for the data analysis to work properly. The SIMBAD CDS catalogue lists is as V=6.21 (from the Ducati [2002] photometry catalogue) and that is also its magnitude in the Yale Bright star catalogue. Of course, it is a variable and has been much brighter in the recent past, so was a naked eye object and it currently is a naked eye object. The system has a binary nature, though the secondary companion is of much lower mass and contributes only a small fraction of the light (Mehner et al. 2010, ApJ, 710, 729).
Allen & Hillier (1993, PASA, 10, 338) give a distance of 2200 +/- 200 pc using a so-called "expansion velocity" method. The star may be part of a larger association that includes the Tr 14 and 16 clusters that have a distance of 2900 +/- 300 pc (Hur et al. 2012, AJ, 143 41). So probably it is more distant than the 4 objects I listed above and is a current naked eye object (late 2014) with V$\simeq 5$  (see http://etacar.fcaglp.unlp.edu.ar/plots/historic.jpg).
An issue in "seeing" Eta Car with the naked eye is that it is mixed up in a lot of bright nebulosity, so it is hard to say to what extent you are seeing the star or the star plus a lot of its surroundings - see below for a Digitized Sky Survey image - messy!

A: The Extended Hipparcos compilation (Anderson+, 2012) provides distances for stars in the Hipparcos catalogue, together with the apparent magnitude in Johnson V (which approximates to how bright a star appears to be to the human eye).
The distance calculation is Dist = 1000/Plx * (1+1.2*(e_Plx/Plx)^2).
Applying the search criteria Vmag<6 & Dist>0 (i.e. not null) identifies 4 stars in Carina that are over 2450 parsecs (over 8000 light-years), all of which are "single components", with some degree of variability:
HIP 51623=HR 4144            2497pc Vmag=5.98 (var < 0.06mag; Hpmag = 6.075)
HIP 52405=HR 4198 (V519 Car) 2492pc Vmag=5.36 (0.06mag < Var < 0.6 mag; Hpmag = 5.446)
HIP 51192=HR 4110 (V399 Car) 2484pc Vmag=4.65 (var < 0.06mag; Hpmag = 4.778)
HIP 52004=HR 4169 (V370 Car) 2475pc Vmag=5.47 (0.06mag < Var < 0.6 mag; Hpmag = 5.47)
Depending on how reliable the distances listed are, any of these might be worthy candidates.
