So, what does our immediate stellar neighbourhood look like? NB: Initially, I chose twelve light years as the maximum distance, but then I got more ambitious.

Alpha Centauri: Ternary star system consisting two sunlike G stars and a red dwarf, Proxima Centauri.
Thus far: Proxima Centauri has two identified planets, with one additional one suspected.
Alpha Centauri A has one identified planet. 
Distance: 4 light years

Significance: Rann is in this planetary system in Earth-One's universe.

Barnards Star: Red dwarf star
Thus far: One-time suspected planet, existence currently refuted
Distance: 6 light years

Luhman 16, WISE 0855-7144
Closest brown dwarves to our solar system.
No detected planets
Distances: 6.5, 7.4 light years

Wolf 359
Red dwarf
One identified planet (intra-mercurian, sunhugger). A second, more remote one
may exist, also outside the star's habitable zone.
Distance: 8 light years
Suggested names: Icarus, Daedalus

Lalande 21185
Red dwarf
Two identified planets (intra-mercurian, sunhugger). A second, more remote one
also exists, also outside the star's habitable zone
Distance 8.3 light years
Suggested names: Chang, Eng

Sirius
Large white A star
White dwarf companion (Sirius B)
No detected planets
Distance: 8.7 light years

Luyten 726-8
Variable binary stars
No detected planets
Distance: 8.72 light years

Ross 154
Variable red dwarf star
No detected planets
Distance: 9.7 light years

Ross 248
Variable red dwarf star
No detected planets
Distance: 10.7 light years

Epsilon Eridani
G type main sequence star
One detected planet (Aegir), another suspected, as well as two asteroid
belts. Aegir is either an ice or gas giant, outside Eridanian habitable zone.
Distance: 10.47 light years

Lacaille 9352
Red dwarf star
Two identified planets, another suspected. Both are superearths and 
both are intra-mercurian. However, a third world may exist within the
star's habitable zone.
Distance: 10.7 light years
Suggested names: Atropos, Lachesis, Clotho

Ross 128
Red dwarf star
One identified planet- Earth-sized, but intra-Mercurian orbit, making
life highly unlikely.
Distance: 11 light years
Suggested name: Inferno

EZ Aquarii
Blue and white ternary variable stars
No identified planets
Distance: 11.1 light years

Procyon 
Binary A and white dwarf stars
No identified planets
Distance: 11.4 light years

61 Cygni
White A star
Possible binary, with brown dwarf companion
One identified planet
Distance: 11.4 light years
Suggested name: Ariadne

Struve 2398
Binary red dwarf stars
Two identified planets, provenance uncertain.
Distance: 11.5 light years
Suggested names: Bacchus, Minerva

Groombridge 54
Binary red dwarf stars
Two identified planets, possibly both ice giants (innermost 
'hot Neptune')
Distance: 11.6 light years
Suggested names: Mephitis, Tyche

DX Cancri
Variable red dwarf star
No identified planets
Distance: 11.7 light years

Epsilon Indi
Binary G and M stars
One identified planet- Jovian size, orbits at equivalent distance to Saturn
Distance: 11.8 light years
Suggested name: Odin


Tau Ceti:
G type main sequence star
Four identified planets, a further four suspected, asteroid belt:Sizes uncertain,
but may be Earths or superearths
Distance: 11.9 light years
Suggested names: Vulcan, Apollo, Phosphorus, Hesperus, Lucifer, Georgium Sidus, Herschel, Leverrier

Gliese 1861
Red dwarf star
Three identified planets, all of which orbit around habitable zone, either earth
or superearth sized.
Distance: 11.9 light years.
Suggested names: Deianira, Hercules

After this, I've only provided details for exoplanet-laden planetary systems. Again, the fact that unplaneted stars do not have currently detected planets is no guarantee that more precise telescopic imaging technology may not verify or refute their existence at some point.

Luyten's Star
Red dwarf star
Two detected planets, two candidate (ice giant/superearth) planets further out. One such world
is in its habitable zone and may have liquid water. The innermost planet is tenuous and may be either
a hot Jupiter or hot Neptune.
Distance: 12.3 light years
Suggested names: Ra, Isis, Thoth, Horus

Teegarden's Star
Red dwarf star
Two detected planets, within and on the outer fringe of its habitable zone.
Distance: 12.5 light years
Suggested Names: Lakshmi, Vishnu

Wolf 1061/Gliese 628
Red dwarf star
Three detected planets, probably two terrestroids/superearths and a third, more remote 
superearth/ice giant/gas giant. Wolf 1061b may be within its star's habitable zone.
Distance: 14.1 light years
Suggested names: Baal, Anath, Moloch

TZ Arietis/Gliese 83.1
Red dwarf star
One detected planet, orbiting at about 2 AUs, slightly more distant than Mars in our solar
system.
Distance: 14.5 light years
Suggested name: Bellona [Mars' consort]

Gliese 687
Red dwarf star
Two detected planets, both ice giants. Gliese 687 b orbits within its stars habitable zone
and might harbour large satellites capable of supporting life
Distance: 14.8 light years
Suggested names: Scylla, Charbydis

Gliese 674
Red dwarf star
One detected planet, either a hot sub-Neptune ice giant or superearth. Intra-mercurian orbit.
Distance: 14.8 light years
Suggested name: Amphitrite [Neptune's consort]

Gliese 876
Red dwarf star
Four detected planets, but while the innermost is terrestroid in size, it may be too close to its star
to be habitable. The other three are Jovian and superjovian gas giants and an outermost ice giant
Distance: 15.2 light years
Suggested names: Hinenuitepo, Papatuanuku, Ranginui and Maui

Gliese 1002
Red dwarf star
Two detected planets, orbit within habitable zone and may be either terrestroid or superterrrestroid.
Distance: 15.8 light years
Suggested names: Montague, Capulet

Gliese 832
Red dwarf star
One detected planet, jovian and orbits at distance comparable to Jupiter. However, even if it does have
large satellites, its distance from its small  primary precludes habitability.
Distance: 16.2 light years
Suggested name: Hera.

Gliese 682
Red dwarf star
Two disputed planets, both within habitable zone.
Distance: 16.3 light years
Suggested names: Frater, Soror (Latin: Brother and Sister)

40 Eridani
Ternary system: A (a main sequence orange star), B is a white dwarf and C is a red dwarf
One planet, in intermercurian orbit
Distance: 16.3 light years
Suggested name: Charis

Gliese 3323
Red dwarf star
Two planets, one in potential habitable zone.
Distance: 17.5 light years
Suggested names: Ishtar, Erishkegal

Gliese 251
Red dwarf star
One planet, a superterrestroid
Distance: 18.2 light years
Suggested name: Demeter

Gliese 229
Red dwarf star, with brown dwarf companion
Two planets, both akin to Saturn in size. The innermost lies
within the star's habitable zone and may have large satellites that 
could support life.
Distance: 18.8 light years
Suggested names: Orpheus, Eurydice

Gliese 754
Red dwarf star
Single planet of unknown size
Distance: 19.2 light years
Suggested name: Rorshach

Gliese 752
Binary red dwarf stars
One planet, with an orbit partially within the habitable zone. Neptune analogue. Again,
may have semi-habitable large satellites
Distance: 19.2 light years
Suggested name: Posiedon

Gliese 588
Red dwarf star
Two planets. Provenance unknown
Distance: 19.3 light years
Suggested names: Aphrodite, Ares

82 G Eridani
G main sequence yellow star
Six planets. Outermost orbits at near-earth distance
Distance: 19.7 light years
Suggested names: Hermes, Iris, Hephaestus, Hades, Kronos, Primavera

Delta Pavonis
G yellow giant, about to begin transition to red giant phase.
One planet, Jovian class, orbiting at Saturn's distance
Distance: 19.9 light years
Suggested name: Zeus

SIMP JO13656.5+093347
Brown dwarf/superjovian rogue planet
Distance: 19.9 light years
Suggested name: Dis




As you can see, I've stuck mostly to the old Greco-Roman planetary naming conventions, apart from Tau Ceti, which has the names of all of the hypothetical planets and alternate planetary names for Mercury, Venus, Uranus and Neptune in our own planetary system. Of course, if it turns out to be China or India that launches the first starships instead of Europe or the United States, the eventual names should be quite different. 

I compiled this list on the basis that some greater detail of the local immediate stellar neighbourhood might be useful to other 5 Earths authors if you want to include it in local star charts, particularly in the context of starfaring metahumans like Superman, Supergirl or members of the Green Lantern Corps.  Interestingly, Alpha Centauri is the only planetary system identified within Earth-One's continuity and seems to be quite divergent from the depiction of Rann as an earthlike planet in an orbit comparable to Earths (although probably at a greater distance, given the size of Alpha Centauri A and B compared to our own sun). Perhaps the internal configuration of the Alpha Centauri system differs considerably from that of Alpha Centauri in our own universe, though?

I think the question now is, what about the universes of Earths Two, Three, Four, X and S? The Justice Society seems to have kept their adventures earthside for the most part, although I wouldn't put it past the Crime Syndicate to conduct interstellar raiding missions, particularly Ultraman or Owlman, if he thought there was something that might benefit the Syndicate out there. The Freedom Fighters and Squadron of Justice seem to have similarly kept themselves earthside. As for Earth-Four, Captain Atom is probably the only one with potential interstellar capability and has certainly contacted alien lifeforms beforehand. 

Source: en.wikipedia.org/wiki/List_of_nearest_stars_and_brown_dwarfs