@Andrew, @Spartacus
I agree that it would be unpopular with Google and Facebook.
But copyright is not the only way to protect a user's data. A hypothetical Pirateland could do away with copyright and still have strong Data Protection rights, after all the "right to be forgotten" is not a copyright issue.
And if we do it through property rights, then whatever the economic value of a user's data¹, the user would have to pay that price to use Facebook or Google. So there's net economic gain to them. It's free Netflix for the price of Google knowing you need incontinence pads and everybody being able to see your photos. Whether that's economically rational depends on the price you put on your privacy.
¹ I can't find the article where you calculate it for an American, but Facebook has quarterly revenues of $1273E6 and 1550E6 users so it generated $0.82 per global user in one quarter of 2015.