Well, as a matter of fact, if well done, it's a quite good idea. For starters, you should NEVER use a random picture. Instead, let the user create an album of rotating profile pictures to avoid showing unwanted pictures.
With this, you can use some nice tricks to give your site a knack. For example, create a profile picture competition where your users can invite their friends to vote for them, and their friends can sign up looking to earn the price, expanding the network with other friends and so on. Offer some cash price, create some decent voting procedure, as transparent as possible and voila. and just in case you're wondering, people may show their looks or create incredible still motion profiles or add some funny stuff. If you're smart and lucky, you might get some viral promotion as well.
And if you don't want to go the above path, you could simply give the feature as a reward. Or create a beta, and test everything (which you should, anyways) .
Bottom line is: try to innovate. Everybody and his mother would love to create the next Facebook. By copying it, you may get a decent copy of Facebook at most. By definition. Think about this: nobody would have give a cent for a social media site offering to restrict you as much as possible. Yet Twitter did it.
Innovation, personality and distinctive features often surpass "doing the right thing", because to create something, you have to go BEYOND the existing and accepted ways. But be careful: breaking the rules without knowing them first is usually a failure waiting to happen. The secret is to break the rules in the exact point they need to be broken.
In short: don't be afraid of innovation, just don't go "blind" and test as much as you can.