Alla en el Rancho Grande (Lorenzo Barcelata, 1925)
When It's Night Time in Nevada (Pascoe, Dumage, Clint, 1931)
I'm a Texas Cowboy (trad.)
Keep Smiling, Old Pal (trad.)

Most of my favorites are the REAL cowboy songs, from before the radio/movie/rhinestone cowboy days, but they're the usual suspects, already listed. Elsewise, my favorites are (false modesty aside) old cowboy-era poems I put tunes to myself, like:

Batchin' (Charles Badger Clark, 1915)
Coyote (Bret Harte, 1869)
Cowboy's Wild Song to His Herd (Wesley Beggs, 1912)
Old Buck's Ghost (Frank Benton, 1903)
Disheartened Ranger (M.B. Smith, ca. 1874)
Night Herding Song (Harry Stephens, 1909)

The Carter Family sang an uninspiring version of "Cowboy's Wild Song to His Herd," and you can find several tunes to "Night Herding Song"--notably, Don Edwards does it--, but I'm unaware of previous settings for the other poems. I've set some others, but for me personally, they're second-stringers, not as likely to burst forth from the shower unless I'm trying to refresh my memory of them. Yup, that's my criterion for favorites: which are rusty from the shower?

I used to groan at "Bury Me Not on the Lone Prairie" till I ran across a great old tune for it in a minor mode. Now I sing it interleaving the two tunes, and it's become a shower song. (Of course, by the time I'm finished, so is the hot water.)

I've also come to enjoy "Home on the Range," "Red River Valley" and some other songs I was conditioned to snub as a boy--they're really quite nice once you get past the done-to-death bits.