SPOILER WARNING!
It has been a few years but I seem to remember at the end the Court awarded the baby to John Wayne instead of Laawman Ward Bond and his wife even though they were actually next of blood related kin. How could the Court do that instead? John Wayne was reformed but still a convicted outlaw while Bond was a respected laawman and real kin. A lawyer today could go to town on this.
I don't know how it would go over in court these days, but in the movie they were honoring the mother's dying wish and Hightower's solemn promise to a dying woman.
Nowadays, being a godparent is pretty much lip service, but back then, when honor and a man's word meant something, it was a real responsibility. The godparent promised to help to bring up the child in the church, and in the case of the parents' death to step in as not only spiritual but legal guardian. With this interpretation of "godfather" in mind, Hightower's claim took precedence over any other, all things being equal. The judge was only waiting to see proof of Hightower's moral fitness before he made the judgement his heart told him to make all along.
Keep in mind too that Arizona wasn't even a US Territory until 1863, and only became a state in 1912.
The movie takes place some time in the 1880's, based on Yuma being a prison and the rail road being present. I don't know much about territorial law, but it would make sense that Wayne had a "claim" to the child, possession being 9/10ths of the law likely extended to minors too.
The only reason there would have even been a judge and hearing was because the town was trying to get established, I think in reality if "Welcome" had existed, it would have resembled "Deadwood."
"if it was any good they'd have made an American version by now." Hank Hill