I'm no psychologist, but here's the first three things that come to mind on what we could enable online to bring current offline-only behaviours online:
1) Groups & Identity Management: This is one that Google+ users are rather annoyed at and has enough discussion on it already. I obviously have multiple groups that I am in, and each of them has a different impression of me: my dance friends don't really care about my job, and my android friends don't really care about dancing. I need a way to intuitively maintain separate personas (either completely public or partially public) and be able to share and discuss within those personas without spamming people in the other persona with stuff they don't care about. These identities need to also be cross-platform: People don't have a Facebook persona and Google persona, they have a business persona and a dance persona and (whatever else your interests might be).
2) Natural discovery: Little kids learn by experimenting and discovering. If you encourage a bunch of kids to learn, they will do so out of pure enjoyment. If you force them to learn, they will hate it. And what happens when they grow up? Well, scholars learn by "publish or perish". Modern online services are directed at tailoring themselves to particular niches, and targeting groups. The key here is targeting and coersion – the opposite of encouraging the natural discovery that gives children subject to that kind of learning a significant advantage over their peers. Rather than tailoring online tools, why not design encouraging ones?
3) Plug-and-play Networks: There are some services that are more or less plug-and-play: Paypal's payment button, for example, can be universally embedded without the user having to understand much about technical code. But how about introducing two groups of people to one another – sort of like everyone does when they hold a birthday party or a farewell party? This shouldn't be something limited to those with online savvy and the 'super-connectors' and 'ringleaders'.