We don't know anything about the Klingon etymology of the word. 'eQway (without the final qaghwI') means "bellybutton". way' by itself means "to parry". We don't know of any word ??'eQ??.
In real life, there is speculation that Dr. Okrand may have been creating a pun referring to the fact that a computer mouse helps you located yourself on the X and Y axes.