What is the word for inventing a new word?

Inventing a word is called "coining", in likeness to the way a government coins money by stamping out metal disks.

The expression "to coin a phrase" means to invent a new phrase, but it is most often used humorously to excuse the use of some very common, overused phrase.

A new word, especially one that is starting to get used but is not yet universally accepted, is called a neologism.