Facebook seems to be losing sight of what people want to use them for: snooping on their friends. ;) I wonder what the MVP of a Facebook de-throner would be. Google+ showed pretty well it wasn't about fighting them head-on. Instagram was making pretty good progress in an oblique fashion. What other directions could you come at this problem from?
IMO the competing MVP is Twitter stripped down as much as possible. When I consume my Twitter feed about 50% of all Tweets have a photos or video. So the consumption process is scroll, read just a bit, watch or view something and repeat. For me that pattern is a pleasant experience. Stuff can be removed. The MySpace like backgrounds to start. Thinking big: remove or tone down mentions and hash tags since IMO they're visually distracting. Point is to make a list-based product where people consume photos and videos from a lightly filtered group of peers as easily as possible.
Well, LinkedIn already has the professional identity market locked down, so an entrant would probably want to focus on personal/private identity.
One approach would be something targeted at dating and meeting people who share your preferred activities, while letting you manage how much of your real identity you want to reveal. Think Second Life meets FourSquare, with avatars, pseudonyms and various exaggerated expressions of individuality.
Another approach would be something for sharing information privately with a group of existing friends, building group identity. Think private clubhouse.