I think the fact that the ANH saber was Anakin's is significant. It's Obi-Wan passing on the remaining good from Anakin on to the next generation of Jedi.

Maybe, when Obi-Wan leaves Padme's apartment after confronting her about Anakin's location, we could see her looking down all upset, and then on one of the shots of Obi-Wan after that, we could hear a quiet Force-swoop and see Anakin's saber float over into his robes while Padme's not looking? That way he'll always have the saber with him, leaving it on the Mustafar landing platform when he takes off his robe for the duel and obviously getting it again after (though we don't see him pick up his robe while boarding Padme's ship, strangely enough, he has it on Tatooine).