Hard to say. I love both films, but I have'nt seen "Midnight Cowboy" recently, so I would say "My Own Private idaho". Although the 2 films share the theme of male prostitution, the latter shows that for Mike (River Phoenix' character) it is possible to make love to a man without being paid for it, which he declares in the famous kissing scene with Scott (Keanu Reeves) by the campfire. Homosexuality is shown as a possibility. Jon Voight's character in MC did'nt have that choice. Much had happened since 1969 when Midnight Cowboy came out up to 1991 when Idaho was made.
By the way, in John Glatts Phoenix biography, Gus van Sant, the director of Idaho, reveals that the kiss and the fact that Mike was in love with Scott, was River Phoenix' idea.