Just off the top of my head, here are some examples of queer cowboy names / characters:
Ennis Del Mar and Jack Twist from the movie "Brokeback Mountain".
Hank Steel, a fictional character from the song "Hank Steel, the Real Queer Cowboy" by Dog Fashion Disco.
It's a long bow, but Joe and Brian from the documentary series "Tiger King" were often seen wearing cowboy clothing.
Additionally, some LGBTQ+ country singers who have embraced the cowboy aesthetic include Ty Herndon, Billy Gilman, and Orville Peck. While these names and characters may not be exclusively "gay cowboy names," they represent a range of LGBTQ+ individuals who have been associated with the cowboy archetype in various forms of media and art.