Player accounts, avatars, icons, and friend lists are all planned, and odds are that services like the Armory and even the WoW sign in may be tied to your Battle.net account in some way. Tournaments and rankings will be included (though it remains to be seen how this might be implemented in World of Warcraft), and Battle.net will even offer players the ability to save and share replays of Starcraft II games, and other e-sport-like features.
That last feature would be a perfect fit for WoW's Arenas, of course, but as much as players would like to be able to spectate Arena matches, odds are that there would have to be a lot of behind-the-scenes coding done on Blizzard's side. Starcraft is being built from the ground up to record matches, of course, but WoW's Arena system doesn't have that ability built in, and putting it in might be more work than Blizzard wants to do on that one area of the game. No word yet on when we might get the new Battle.net (odds are it'll come out right around the Starcraft II release, whenever that is), but from the look of it, Blizzard has big plans to expand the social networks they've built into World of Warcraft towards all of their properties.