Green River is a regional soda. It's named for Chicago's bizarre habit of dying the Chicago Rover bright green for St Patricks Day. The soda is primarily available in the Chicago area and is bottled by Canfield's (who also make kickass Diet Chocolate and Diet Chocolate Cherry sodas).
Green River was introduced in 1919.
"As part of a more than fifty-year-old Chicago tradition, the Chicago River is dyed green in observance of St Patrick's Day. The actual event does not necessarily occur on St. Patrick's Day and is scheduled for the Saturday of the closest weekend. The dye takes days to dissipate.
"The tradition of dyeing the river green arose by accident when some plumbers used fluorescein dye to trace sources of illegal pollution discharges. The dyeing of the river is still sponsored by the local plumbers union." - Wikipedia.
So. The soda water was named before the dyeing of the river.
From the nineteen-eighties until 2011, Green River was bottled by Clover Club Beverages. Currently owned by WIT Beverage Company.