This is really hard indeed... I have to say right off the bat that I don't have an instant favorite. Then, let's begin the random pondering!
Actually, Nightgate Inn has proven to be a nice place for me. I've seen that Wandering Bard, the Wanderer (forgot his name) there twice, and both times I've requested a song from him. I've done that very rarely even when I see him out in the wilderness, even though that voice actor has either my second or most favorite voice of any bards in the game. None is miraculous, but few are good enough, and that guy's one of them.
Anyway, those have felt like some V.I.P. concerts to me and that one other customer residing there!
There's something sympathetic in that place anyway... maybe it's the remote location, and the owner's balls to run an inn in a place like that. 
The Bee and the Barb. Riften is basically the hometown of Hides-Swift-Strikes, the other of my two main characters with most hours in, and he's the one who occasionally visits any taverns at all. A good place. Although a thief and an assassin, Hides is still very compassionate for his fellow Argonians and their position in Skyrim. He's always looking to help his marsh-friends - if their needs don't cross those of the TG and especially the Night Mother. The Inn is owned bu such friends.
The Inn also used to be a great place to troll (pickpocket etc.) that [censored], Maven, who thinks so highly of herself. She thinks she owns the TG and DB. Heck, she doesn't even seem to know that Hides is the new head for both of them, and he couldn't be more disrespectful as he doesn't like Maven's attitude. Now who has connections and who doesn't? Why are Maven's belongings constantly disappearing without notice? Is there some rogue thief always robbing her, why isn't the Thieves Guild doing anything...? Haha, she doesn't know [censored], she still thinks her rules count something. She shouldn't have been so nosy to Hides in the first place; now she's paying for it, and she doesn't even know it, since Hides is just too good a thief to be ever caught in robbing from his best client! Some bold old school Bal-Molagmer / Robin Hood stuff over there, the stuff Thieves Guild is famous for!
I said "used to be a great place to troll Maven". Well, not anymore, as she's the Jarl now. Let her be. It's so nice to have a puppet ruler, especially as the ruler seems to think she's not a puppet at all!
What a perfect grasp Hides has over The Rift!
The Candlehearth Hall. Nice bit of lore in the name (overall there should be even more details like this!), and it's the Inn my Stormcloak-Dragonborn Asmund Stone-Singer frequents. The 3rd-4th most played character of mine, and the only one who often visits any Inn in the first place.
So I'm kind of undecided between those. The one in Riverwood also has a special nostalgic place in my heart, since the start of the game really impressed me on 11/11/11! 