I've always a) known this sentiment as the Golden Rule and b) thought it was so-called because it was a basic simple rule for behaviour which appears to transcend any one religious/ philosophical culture or background. Analogous with the 'Golden Ratio' in art. (The ratio between any two consecutive numbers in the series 1:1:2:3:5:8:13:22...(each number being the sum of the two previous) - the more precisely you locate this in the ratio, the 'better balanced' the picture.)
I'd be willing to bet (a small amount!) that the preponderance of Biblical references is at least in part a reflection of the 'Anglo' ethnocentricity of the web and many reference sites on it rather than the origin of the sentiment itself.
So why 'golden'? (I couldn't find an origin on the web quickly either!) Well, gold is a valuable thing across a vast number of cultures too - and as an inactive element one of the few metals that occurs naturally looking bright and shiny and attractive. And stays that way. So lots of connotations of good, valuable, true etc. But hey, I'm an amateur, speculating wildly. No sources quoted as I have none!
