A noun in The Etymologies of the 1930s given as N. rhim “cold pool or lake (in mountains)” derived from the root ᴹ√RINGI “cold” (Ety/RINGI). This word is the final element of N. Mithrim “✱Grey Lake” in The Etymologies, but in later writings Tolkien explained this name differently, deriving the name from the Elves that lived around the lake, so it may have been abandoned.
Neo-Sindarin: If adapted to Neo-Sindarin, this word would be ᴺS. rim. Since √RINGI “cold” survived in later writings, this word may still be viable, though it would conflict with S. rim “host, a great number”.