This is much more than just a petition, though, even compared to a White House petition, which still has almost no chance of solving anything, other than receiving a patronizing and vague response, if that. The proposals in this "petition" get on the floor in Parliament it seems (if it passes the 50,000 mark), and they have to vote on it.
I really like the idea, and I wish more countries adopted this. If you're not a big fan of direct democracy and citizen-referendums like in Germany, this one still allows the representatives to have the final say on it. So nothing changes from the republic point of view, but it does offer regular citizens much better access to the Parliament and getting it to vote on laws they care about.
Of course they can still vote no, and they might, but that's why you need to build enough awareness and also try to change the politicians' mind on the issue.
or if its a badly worded motion/petition it would have to pass the rules for validity according to the rules of the finish parliament and if it failed that it wouldn't even get on the order paper let alone moved.
In Brazil, a new law can be proposed by regular people if it gets one million signatures. While congress doesn't have to vote these proposals (they can basically postpone voting them forever), some of these have indeed become law.
The most recent one was proposed in 2009 and became law in 2010, changing the requirements to run for office (making them somewhat tougher on corruption).
PS: That law was nicknamed "Ficha Limpa". Something like "Clean Slate".