The nen- element of this river's name is definitely 'water', a very common element in Elvish river-names. The ending -ning is obscure, but perhaps derived from ninn, meaning 'slender, narrow'. If this conjecture is correct, then, the name Nenning meant 'narrow river'. This idea is supported by the fact that the Nenning was known to run through a ravine, at least towards its mouth.
In earlier texts of the Silmarillion, this river is known as the 'Eglor'. Tolkien first created the name 'Nenning' while writing The Lord of the Rings, where it was one of a sequence of proposed names for the river ultimately named the Celebrant or Silverlode.