Actually now in britain we have it set up so whenever one of these gets 100,000 votes, it must be debated in parliament. One of the few useful or interesting things cameron has done. Of course they are quite rightly not bound to do anything more than debate it, and in the long run it will probably just be a more efficient way of shutting people up. "Look, we've debated it, we've given parliamentary time to your issue, what more do you want? This is a representative democracy you know!"
fuel prices - http://epetitions.direct.gov.uk/petitions/347 (small decrease implemented, plans for the future shared)
immigration - http://epetitions.direct.gov.uk/petitions/19658 (basic plans shared, recently implemented changes listed, already heard some people having visa issues, so something is happening)
summer riots - http://epetitions.direct.gov.uk/petitions/7337 (no direct actions, but response means pretty much "this is happening anyways", there's some more information about what the local authorities can do)
Hillsborough 1989 - http://epetitions.direct.gov.uk/petitions/2199 (accepted, disclosure will happen)
financial education - http://epetitions.direct.gov.uk/petitions/8903 (this one does look like a non-answer, but maybe they're really doing the reviews now)
private pensions - http://epetitions.direct.gov.uk/petitions/1535 (no response yet, only crossed the line recently)
While I'm not crazy about all the answers, there was some effect (or at least an explanation or a commitment to the original plan). They're quite far from "a more efficient way of shutting people up" the way it's presented in the comments here.
When we get a million signatures against war in iran, or to unilaterally leave the EU or something, then we'll see what happens.
Even worse, if enough petitions get that many votes, it could be spun as a problem for politicians who want to get things done, "The public keeps wanting us to debate these stupid issues! We can't get to the more important things!" Yes it sounds far-fetched, but some newspapers will sink that low...
In the truth-vacuum of DC, it's pretty relevant.
OT, but is it then reasonable to infer that the increasing closeness of Presidential elections over the past couple of decades may have simply been a cost-cutting operation by the lobbying industry?