The online magazine Slate reports that Finland recently launched an open-source web platform called Open Ministry to allow citizens to propose legislation, which must be voted upon by Parliament if it receives the online attention of 50,000 citizens.
Each suggested law gets six months to gather traction. Whether the majority is in favor or not doesn’t matter, as anything with 50,000 total shares (likes or dislikes) moves on to the next, official round of voting. Two weeks ago, a proposal to ban the practice of farming animals for the fur trade became the first Open Ministry idea to pass the threshold for Parliament[ary] consideration. Out of the roughly 340 pitches currently on the site, the fur-trade idea is far and away the most popular, having collected more than 56,000 shares with the majority in favor of the ban.
The closest federal analog in the U.S. is Rep. Issa's innovative Madison project (code available here) that allows citizens to comment on legislation online and see each other's comments -- all of which is available for review by the Committee on Oversight and Government Reform. The Finnish example goes further because (1) citizens can propose legislation, and (2) sufficient attention for legislation requires parliamentary action.
In the federal executive branch, there seems to be an indirect comparator to Open Ministry in the notice-and-comment rulemaking process that agencies must follow, whereby they're required to respond to public input before promulgating rules and publish their responses in the Federal Register. In a different sense, the White House's citizen petition platform "We the People" is also related, as enough votes triggers a public letter in response to citizen inquiries.
It would be interesting to see Congress incorporate the White House's citizen petition platform into how it responds to constituent letters, much along the lines of what's happening in Germany with Parliament Watch, where responses are available for everyone to view.
I'm not sure it would make sense for Congress to adopt the Finnish model wholesale, however, as their parliament and our legislature are very different institutions. But it is conceivable that a committee could decide to introduce legislation if a certain number of petitioners indicate their support for a measure, and then take whatever action is appropriate. A committee could even agree to take certain measures up for a vote before the full committee, although that may run into practical and prudential limitations.