CLEVELAND—Len Olds and Hugh Rouse went to their first California Republican Party convention in 1982, when passersby mistook their gay-rights brochures for sex-club ads. They attended their first national convention in 2000, cheering then-Rep. James Kolbe of Arizona, an openly gay congressman, for speaking over the objections of the Texas delegation.

The gay couple and longtime members of the Log Cabin Republicans, the largest gay GOP group, feel more optimistic about their party than ever this week at their fourth national...