The Line River is a no-longer-existent river that used to run along the lower edge of the Sandara Desert, supporting the agriculture of the Sand Kings civilization with its annual flooding. The river was clogged with debris in The Cataclysm and filled with sand in the Sandstorm of Apep. This changed the course of the river, which eventually dried up completely.
The river was given an official name after it had dried to a riverbed, by a not very creative explorer who observed that the river looked like a line on the map he was shown by his guides. "Of course it does! It is a line. That's how we draw all rivers," his guides told him. But he didn't listen.