Does 青龙 not really mean Green Dragon? People at the Hebei museum, and my friends, and Baidu http://baike.baidu.com/view/606163.htm all agree that 青龙白虎 means "Dragon and White Tiger."  (I capitalize because these name specific folk animals.) When I asked at the museum whether it means the dragon is green they could not see why I would ask that.  So I suppose 青龙 is a traditional name that just means dragon and not green.

Can anyone here tell me if that is right? And can anyone explain a reason for it?  Or is it just the kind of thing can happen with traditional names in any culture?

Actually, on line a lot of people translate it as Azure Dragon.  I know 青 has several more meanings than the western green/grun/vert.  So maybe just plain Dragon is not a good translation of 青龙.  But I do not know.
青龙 should be translated as 'Azure (or Green/Blue) Dragon', that is the accepted translation for the mythological context. (Check some dictionaries, Pleco or CEDICT, they all use Azure Dragon, Wikipedia, and this Wiki article on the Four Symbols, too.)

Why would 青 mean nothing here? You can argue if it is blue, green or azure, but it is a color. Perhaps the people who translated 青龙白虎 as 'Dragon and White Tiger' just got confused about the different equivalents of 青 (blue, green, black) and left it out. 

If Wikipedia is not authoritative for you, you can try a site colon search on Google with site:edu 青龙 . This will search for 青龙 on the websites of American universities. You can see that most of these sources will give the translation as either Blue, Green or Azure Dragon. 

If you are still looking for some evidence, there are some books written on (or inspired by) Chinese mythology with the title bearing your expression: