this is one of those cases where i just want to ditch the established names and just go with what makes most sense to me, but i'd like to be able to understand when other people write music too.
i would recommend calling it Gm7b5 or G half diminished. i suppose the reason it's called half diminished is because the 5th is diminished and the 7th is not. i think in the traditional view, for major/minor intervals, 'diminished' actually means an additional halfstep flat. so a diminished 7th actually technically is the same notes as a major 6th. none of this is really important unless you like it