Aggregators would be where you go to set up your friend list and see your feed. It could look and feel like Facebook does now. It would have an open standard protocol that content hosters would use if they wanted to be aggregated. This could still be an add driven business, but subscription, self hosted, and DIY solutions could exist too.
Content hosters could either charge a monthly hosting fee, or they could serve up their own adds. Self hosted and DIY solutions could also exist.
The big benefit to this would of course be the competition. Since it's an open standard anyone could be a content host, and anyone could be an aggregator.
To make extra sure there is competition, and this could come in a phase two after the initial splitting up of Facebook, there should be open standards for exporting and importing friends, follows, likes, etc. to and from aggregators, and open standards for importing and exporting content from the hosters.
Speaking of follows and likes, there could also be aggregator aggregators (AAs). People could opt in to publicly and anonymously share their likes and follows and the AAs would consume those and report on trends that cross aggregator boundaries. Anonymity could be much more protected this way while still giving us that interesting information about what is trending.
One tricky part of this is how do I as a content author only allow my friends to see certain posts of mine? It would have to be with encryption. My content provider could keep public keys of my friends and only my friends (well, their aggregators) would be able to decrypt my posts using my friends' private keys. I can see some challenges and holes in this, but it doesn't seem any worse overall than how Facebook protects privacy now. Open implementations and peer review could get us to better-than-Facebook privacy quickly.