A little internet research and I have found some answers. There was an actual bear named Winnie, a black bear. She was purchased by a Lieutenant in the Canadian military prior to shipping overseas to Europe during World War I. The man named the bear "Winnie" after his hometown of Winnipeg, Manitoba. Winnie was a pet and Brigade mascot until the troops actually had to start fighting, upon which he was donated to the London Zoo.
The author of the Winnie the Pooh books, A. A. Milne, had a son named Christopher Robin who was enamored of the bear at the zoo and named his own teddy bear after Winnie. The name Pooh comes from a swan that A. A. Milne had previously written about. A. A. Milne was inspired by his son's relationship with his stuffed animals, and so he started writing his Winnie the Pooh books.
So really, I still don't know why anything, swan or bear, would be named Pooh. But it's a pretty interesting little story that I knew nothing about until tonight. I have a lot more respect for the little bear. And don't worry, I didn't spent that much time researching this.