I don't get why Stern would want Miami to win. I understand why people would think he wants Miami in the Finals, but the NBA is going to get a lot more attention this offseason if the Spurs win. There will be tons of speculation on LeBron's legacy and the future of the big three in Miami.
If the Heat win though, it's just another notch on LeBron's belt and solidifies the Heat as a dynasty. In terms of getting attention for his league during the summer and fall, it's a no brainer that he would want the Spurs to win.