The Economist has an excellent survey article, here is one (not the only) estimate:
Yet another technique is to assign a value to the leisure time spent on the web. Erik Brynjolfsson and Joo Hee Oh of the Massachusetts Institute of Technology note that between 2002 and 2011, the amount of leisure time Americans spent on the internet rose from 3 to 5.8 hours per week. The authors conclude that in so far as consumers must have valued their time on the internet more than the alternatives, this increase must reflect a growing consumer surplus from the internet, which they value at $564 billion in 2011, or $2,600 per user. Had this growth in surplus been included in GDP, it would have raised economic growth since 2002 by 0.39 percentage points on average.
I would note one caution. Consumer surplus per se does not make published gdp figures inaccurate for most purposes, since all goods and services yield consumer surplus to some extent. One might argue, however, that the internet has higher than average consumer surplus, for purposes of thinking about human welfare.