I was discussing the idea with a friend of mine, and we came to the conclusion that having a personal base setup is a good idea.
The big thing is that a lot of heroes have their own personal base of operations.The examples that spring to mind for me are Superman's Fortress of Solitude, the Batcave, Peter Parker's house, Iron Man's workshop, the X-men's school, and innumerable villain bases. The big thing with these is they put in a locational identity to a given character, which would help you out from a roleplaying angle quite a bit, and gives people the chance to do their "an interior decorator is you" thing. I know some people back in coH lamented the inability to have a personal abode, and I almost certainly would have put together a small apartment for a few of my characters.
Ignoring the fluffy bits for a moment, there's a potentially a lot that can be done with a personal base mechanically.
First and foremost is trainers - back in CoH, locating and getting to trainers was a hassle, but I still enjoy the essence of going to a trainer more than just auto-leveling wherever you are as seems to be becoming a more common thing. Imagine this - you've leveled up after a recent Task Force run, but leveling up through a trainer tweaks your brain because it doesn't fit in with what you imagine for such a character advancement; maybe your character works with unique tech that he developed for himself, and his way of improving is optimizing his tech or making new gadgets. Maybe she has a pact with infernal powers for... well, power, and goes home to summon up an entity she can talk with to upgrade her contract. Maybe it makes the most sense for him just to go home and take a nap, and growth happens naturally while he's asleep. Maybe she has a gym she spends a while training his abilities in. If you can customize what your personal trainer looks like, it opens up a variety of things you can do that make sense in character. Having that option for character customization is a big bonus.
But then we can also start looking at other services you might want to be able to access from home. The big ones that come to my mind are the costume shop, the auction house, and storage. Storage should be largely self-explanatory and doesn't deal heavily with stuff like banks, which are kinda weird for taking items from a fluff point of view (in Neverwinter, I have a horse sitting in my bank. Really?) but make sense if you're just storing your stuff at home, in closets and boxes or even in the back yard or basement.
The auction house and costume shop are a bit trickier, but I'm sure you guys remember having to cross multiple zones trying to remember where the hell the costume shop was, or running along a dangerous road while underleveled to get to the costume shop on your lowbie character, but this makes sense if you think about it. Accessing an online exchange of goods from the computer is something that happens a lot nowadays; a friend of mine even has her groceries delivered to her home, since she can't leave often due to health issues. The same concept applies for costume and auction house shopping, as both can be theoretically available from home. Maybe your character has a computer that he buys things to be delivered off of. Maybe you have a demonic auction broker you can summon up. maybe you tinker with your gear and "make" new ones using blueprints you found online. Maybe you just have a really, really extensive wardrobe.
"But Halae," you may be asking, "How does any of this matter if the home is in such an out of the way location that you never go there?" Well I'm glad you asked. the thing is, we're going to have travel powers; that was guaranteed, right from the concept phase of the game, since it's based on super heroes. I, personally, am imagining that many different power sets would have a quick "return home" power available. Teleportation could take you right there, flight could could be a fly off into the sky and vanish animation. Super speed you just leave very quickly, and a vehicle power could have your vehicle arrive, you get on, and drive away. Cue loading screen regardless of the power in question, and you're instantly at home. This is the sort of thing that could even be earned ingame, rather than needing to spend points on it, since it's more convenience than an actual power.
Anyways, this sort of system could have quite a large impact in character (i know where my character lives), mechanically (I've got lots of amenities at my character's home) and out of character (players love having their own house and such in MMOs), so I feel it's worth looking into.
An infinite number of tries doesn't mean that any one of those tries will succeed. I could flip an infinite number of pennies an infinite number of times and, barring genuine randomness, they will never come up "Waffles".