I remember when I was a kid. Several times a year, someone would approach me in the playground or classroom and ask me to "sign our petition". Sometimes they were quite sensible ("Open both doors at the East entrance at lunchtime so we don't get trampled trying to get in or out") but most of the time they were absolute crackpot that only made sense when you were a child ("Get Mr Smith sacked because he gave me an unfair detention!" or "More/larger chocolate desserts for the same price at lunchtime", etc.)
Even back then, I never bothered. Honestly, it just wasn't worth it. You could have the entire place sign the thing and nothing much would ever happen about it, even if the idea was quite sensible (the doors never were both opened as long as I was at that school, for example - I assume there was a reason for this but never quite found out). Petitions really were the playground democracy and, let's be honest, the government will ignore most petitions just like my school did. Even the sensible ones.
It's a gauge for government, that's all. If the country was ruled by public opinion, people would be hung before they were tried, some celebrity would be in charge (until they made their first mistake) and road deaths would increase ten-fold after all the changes people wanted (like to be able to drive like nutters on
the motorway). All the petitions do is give a sense of "contribution", provide statistics about public opinion, but don't actually change anything. If the Jimmy Saville thing had come out earlier, and every person in the country voted to hang him without trial, it still wouldn't have happened. But they can use the list to look at the most-named ones and garner a lot of votes by giving a pseudo-statement to the effect that they'll look into it, and talk about it in the news (because people obviously want to hear about something being done about it, but obviously don't care about the three bills I slipped through the Parliament back door last week).
The largest petition on there attracting over 200,000 names was "Convicted London rioters should loose all benefits." Apart from the bad spelling, this suggests that people who were convicted of a crime should have a punishment not assigned by a court, in a rash legal change, for a single incident only (presumably OTHER criminals are okay, but the wording of the law), etc. etc. etc. And what was the response? No, basically. Of course it was.
The next ten more popular petitions of all time? No, we have already passed the law you didn't want. No (though we talked about it). We did nothing about this (though we talked about it). No, dropping the petrol taxes will cripple the country. We take your point but we can't stop people coming into the country. And, no, because PSHE classes already teach pupils enough financial acumen to survive in the world (really?).
The biggest trending petitions still open are the moment are ALL media-related (West Coast mainline, badger cull, tax at Rangers Football Club, etc.). That should worry more than anything - people care more about things that the news outlets place on their front pages than anything practical or sensible . Lots are inherently misguided. And some are just plain crackpot ("Alopecia Areata - Research Needed" has more names to it than "Save Royal Bolton Hospital").
A petition of any volume WILL NOT CHANGE ANYTHING. All the petitions on that site HAVE NOT CHANGED ANYTHING (and if they were successful, I'd argue they could have been without the petition anyway). If you don't want the West Coast Mainline to change, sure air your view. But the only thing that will actually make any difference is to NOT use the West Coast Mainline if it changes to a company you don't want to support. And that won't even be a government effect, just a purely profit one. The fact is that if it did change, and the government approved it, lots of people would shout for change while still using it every day. You can say "we had no choice", but that just proves how unimportant it is for the government to respond in such cases - they KNOW you have no choice, so there's little point taking your view into account.
It's like objecting to planning applications. Sure, you can. It's there. There's a process, and a form, and a guy, and a meeting that has to happen, and all the rest. But unless there's a REALLY good reason that nobody ever thought of and nobody ever checked and nobody's checklist forces them to consider already anyway, the chances are that your objections will be ignored and overruled. Chances are the number of objections upheld is really quite pathetic, and has more to do with things slipping through the net or personal favours rather than anything to do with "listening to the people".
A petition is worthless. All the ones people have ever pushed into my face have come to nothing. And an electronic one means even less. Of all the government petitions I see for the UK, you only have to get to page 4 of 623 at the moment (20 petitions per page) of the closed petitions before everything goes under 10,000 names. Currently open ones? Page 2. That means that just churning through and responding and administering those petitions is actually causing LESS things to get changed overall than if we didn't have that. We've wasted more man-hours petitioning online and responding to petitions that it would have cost just to carry on as we were and do something ourselves. And the government response to almost every petition? No, or doing nothing, at great expense.
Seriously, if your MP doesn't do anything when you personally write them a direct, open, well-considered, precise letter, what makes you think that an electronic tick-in-a-box does anything for the way they work? It doesn't. It just gives them an indication as to what the best thing to "cover-up" with is at the moment.
What's the solution to actually getting change? I don't know. But a petition is probably the last and worst thing to do.