Well, I can't think of a single one that would be cold hardy in 5.
As to overwintering, its the same as with any plant.

For evergreen gingers (Costus, hedychiums, alpinias, etc) keep it in a pot, bring it in the house before the first freeze, keep it in the sunniest place you can, water it judiciously.

For deciduous gingers (Kaemferia, curcuma, most zingibers, etc) let the go dormant naturally, dig them up and store the rhizomes in dirt or sphagnum in a place where they won;t freeze.

Try gingers from the genus Roscoea. On recommendation from the nusery, I planted them 12" deep and they've survived 2 winters. Mind you, they only grow to 12" tall and don't emerge until late June but they are hardy gingers.

Simon