They tend to grow high up in the hills, so they're associated more with hikers and wanderers than with city folk.    I think also they're probably named in lyrics because of the pleasant cadence of the name itself and because "berry" rhymes with a lot of English words. Anyway, huckleberries are a very "folky" kind of fruit - more so than blueberries or cherries or watermelons and other things you have to cultivate.   

Wouldn't be surprised if Mark Twain's naming a carefree orphan adventurer "Huckleberry Finn" had an influence on later-appearing American folksongs too.