The king of Archenland during the time of The Horse and His Boy, father of the Archenland princes who play a more major role in the story, is called Lune. Is there any significance to this name? Perhaps either a connection with the moon or possibly with insanity? (For what it's worth, Lewis has a history of making symbolic connections to heavenly bodies in his novels, including Narnia.)
Some interesting bits come up on wiktionary: lune is apparently one alternate spelling for lyon, which is one way the leash for a hawk was spelt, as in Sir Thomas Malory:
And thenne was he ware of a Faucon came fleynge ouer his hede toward an hyghe elme / and longe lunys aboute her feet / and she flewe vnto the elme to take her perche / the lunys ouer cast aboute a bough / And whanne she wold haue taken her flyghte / she henge by the legges fast / and syre launcelot sawe how he henge
[or, if you don't want to deal with the middle english:
And then was he aware of a Falcon which came flying over his head toward a high elm / with long [traces/leashes] about her feet / and she flew onto the elm to take her perch / the [traces/leashes] cast over and about a bough / And when she would have taken her flight / She hung by the legs fast / and Sir Launcelot saw how [s?]he hung.
Another meaning, derived from the fact that lune is related to luna, lunar, and therefore the moon: it can refer to the phases of the moon and mean whim, caprice, humor. Another meaning, derived from the Old Norse lugna, means warmth, and to calm.
What struck me immediately was how King Lune is, as King of Archenland, a barrier between Narnia and Calormen. This made me think of him being a sort of leash on both their armies (referring to the hawking definition now), as a protective barrier - and something certainly needed for the diplomatic relations involved in this would be the ability to be warm/charming, and the ability to calm people down (referring to the Old Norse derivative now).
The Keys to the Chronicles: Unlocking the Symbols of C. S. Lewis's Narnia by Marvin Hinten suggests that the names Archenland and King Lune form a linguistic pair - another derivative meaning of the word lune is arch, or crescent - archenland, and a name that means arch. This seems to me a bit of a stretch, but that's one theory.
There actually aren't that many other analyses (at least not that my dig could turn up). I find your bit about the planets interesting, but I'm not sure if the Moon relation quite makes sense - you could argue that Archenland is a bit of a "satellite" of Narnia. I suppose you could also argue that Rabadash goes a bit bonkers and consequently gets turned into a donkey - but that's even more of a stretch in my mind.
Another thought that struck me was that it could be like Cor and Corin - Cor doesn't make a ton of sense on its own, but it's part of his brother's name. Similarly, Lune (or more likely, 'Lun') could be the first part of Lune's brother's name (we do not know, of course, whether Lune had a brother). The only problem with this is that boy's names starting with 'Lun' aren't very common - the main ones I found were Lundie and Lundon (and derivations of each) - the former is Scottish in origin, and the latter an early version of London.
Honestly, it could very well be a combination of these.