[AJames]

Sigma octans is the Bayer designation, but is not a "common name." There is another star invisible to the naked-eye, that is referred to as "A Oct", which was closest to the pole in about 1990, which is about 10.6 magnitude. This has the position of 13h 16m 58s -89d 52' 39" (2000). This was labelled as such in the 1950's as the identifier for the south celestial pole, but its actual origin I unsure of - might be Stromlo or during the AAO field testing?

Another is B Oct (6.5 magnitude), which was closest to the pole until about 1950, but since then is now further than Sigma Octanis. [Neither A or B Oct are designated variables.]

(This forms an important second smaller or littler trapezium of stars, which can be used to visually identify the pole - often required when field-testing for a southern observatory site or precise alignment of the polar axis - taking into account various effects like optical aberrations, atmospheric refraction, flexure, etc..

If you mean "southern named star", then I presume this is as a proper name.

If so, I only know that brightest star is Apus is the yellow Gamma Apodis, at merely 3.8 magnitude, and this was also once known as "Avis", meaning "important" and derived partially from the Latin - but this name isn't very popular. The declination of this orange star -77 deg 30' 02".
Furthermore, Delta (1,2) Apodis (Apus) 16h 20m 21s -78 deg 41' 45' and 16h 20m 27s -78 deg 40' 03", being also the wide double star, BSO 22, is also known to have a name as well - seen as the "red bird's eyes" - seen on many colourful birds - but I can't remember the actual reference. Darn! It is, however, slight south than gamma Aps, which would make it the most southern. (I'll have to find this reference in a more detailed source, though. Sorry.)

As for stars around 3rd or 4th magnitude - certainly Miaplacidus is the most southern. (there are ten other candidates - none that I know of have a proper name. Ie. delta, gamma Mus, alpha Men, beta and gamma Hydrus, beta and delta Octans, alpha Apodis, and epsilon and zeta Pavo.)

Of these a new common name, which I think is apt is Alpha Mensa, which could be named Amen - amen indeed!!

Regards
Andrew


Note: I have two interesting articles at my site on this general subject, if you would would like some general detail; Namely....
"ON the PROPER NAME ‘GACRUX’ and the NAMING of OTHER SOUTHERN STARS" See http://homepage.mac.com/andjames/Page206.htm

and
"GREEK
LETTER DESIGNATIONS of SOUTHERN STARS"
http://homepage.mac.com/andjames/Page208.htm