Sometimes an etymology is so obvious that you can't believe you've never noticed it before. This was true of cappuccinos and Capuchin monks, and is just as true of inauguration.
When you begin something new, it is a good idea to call in a soothsayer, or augur, to see how it will turn out. They can tell you whether today is a good day to start, and whether you should start at all. This is the inauguration.
Augur itself may come from avis, meaning bird, because Roman augurs used to chop up our feathered friends in order to find the future in the belly of a pigeon. Augur may, though, be the cousin of augment because the purpose of an augur was to predict an increase (or augmentation) in crops.
If augur does come from augment, then it relates to author, as an author is somebody who increases and augments the number of a books in the world. Authors have authority, and writers have rights; but bards sit in bars buying dramatists drams.
The Inky Fool thinks it may rain this afternoon.