Releasing a mobile application and want to be able to achieve the same success DropBox did early on. The DropBox approach was giving away free data from user referrals. I am not sure what we could do for a Social Media app
Well, first I have very basic information about this social network. What is it about? The key thing I would recommend is finding your Unique Value Proposition and what do you have to offer your users that they can't get from any other app.
Gamification is a good way to motivate users to share and for other things, like: to earn badges, prizes and so on. Also, having a good valuable social network makes it easier for your users to want to invite their friends to participate.
How do you plan to monetize the social network?
If you have something people have to pay for, trade it for referrals.
If you don't, consider trading "X days ad-free" in exchange for referrals.
If you don't have a revenue stream from users or ads, focus on delighting people and offering an easy way to share after they've just gone through an experience that will make them think, "Holy shit that was awesome."
What dropbox did was in Game Design terms, use the recruitment mechanic. It tells players to gather a team in order to accomplish a goal. "Get together your A-team" as the saying goes. Dropbox also understood, or created the perception, that data stored in the cloud was worth more than on the single device. By giving that currency as an incentive for recruitment, they were able to tap into a reward mechanic of greed.
For social media sites, there are different issues. The early days of Orkut [Google's first attempt at social media sites] became extremely popular in Brazil. Mainly because orkut had a leader board based on the number of members per country. The Brazilians saw that they could overtake America and other countries and it became a viral Epic Challenge for the Brazilians. That's not to say I'm a fan of leader boards, but they tapped into the Glory reward mechanic there.
Facebook tapped into the "God reward" which is all about Access. It was a closed system, only available to certain schools and colleges in the beginning. This meant that you had to be special to be on it. Not like today where everyone including your grandmother is on it.
Look at what your social media app has as far as a USP, and grant more of that for signup. Or allow access only by a certain groups at first. The reward needs to fit your app however, but be strong invite to invite.
I recently gave a talk to National Geographic about how they could increase subscribers by using this mechanic. The main idea of the campaign, "Who would you take on a National Geographic adventure? Fill the seats on the plane, and you could both win the trip." Take the usp and grant access to the experience once more than one friend has unlocked it.
I hope this gets your ideas turning.