Mike Brown. Why would the Lakers pick Mike Brown to be the new head coach?

Well, it’s either one of several things. Firstly, Mike Brown could be the best coach of all time, but no one has realized it yet. Coaching the Cavilers has not helped his reputation as a great coach, but now that he is at the Lakers, that might change. Maybe he could make the team better again, better than they were with Phil Jackson. This is pretty unlikely.
It could also have been a decision that fit with a much larger plan. Jerry West has just been signed to become an executive board member to the Golden State Warriors. This could mean that West is using the Warriors as a farm team, and that he is planning something to help the Lakers. The decision to enlist Mike Brown as a coach could have something to do with this as well.
I still think that the Lakers should have picked Brian Shaw to be the head coach. He has worked with Jackson, and is still young. He even played with some of the older players. But who knows, the Lakers could be planning something big.