Ah, I was thinking it might be the Protomateria because of how Shelke retrieves it, but I decided it wasn't because I'd somehow completely forgotten that Rosso took it from Vincent earlier on. Thanks for reminding me.
As for the "him" line, I kind of feel Weiss is referring to Genesis too, because it seems weird to call Omega a "him", but of course it doesn't really make much sense if he's referring to Genesis. Having said that though, the secret ending in DoC makes me think the creators hadn't hashed out Genesis' story fully when the game was released. It is impossible to see the secret ending as anything but ominous, and yet Genesis is supposed to be a good guy at that point according to CC, which came out later. I wonder if the "him" line is a reference to an earlier version of the Genesis story? Genesis takes Weiss with him in the secret ending too, which could be related in some way to Weiss and Nero's desire to "join" with Genesis. All poor speculation though.