While I’m not yet 18 and therefore cannot vote, I know some people who are apolitical but eligible to vote. Would it appropriate if I managed to convince them to vote one way as a favour for me?
I have done this a lot. It only works for local elections. You won’t be able to influence enough people to chance the results for the national elections.
But…think about this…how did they start? They started small. If you can influence the local elections, you can have a voice in the big ones.
Why wouldn’t it be appropriate?
Well if it’s a situation like, “Qin, I don’t know too much about politics but I trust you’re judgement” then I think it’s A-OK.
If it’s - Qin:“Come on! just do this for me - I’ll owe you one, you need $50 to get your guitar out of hock, right?” then not so much.
I guess it depends on how you define “a favor”.
Only if you convince them to vote Democratic. 
By that argument, shouldn’t one not vote in national elections at all? Casting one vote yourself and convincing one person to vote for what you want has the same outcome in number of votes.
You can persuade people to vote. You cannot give them money to vote. Money isn’t speech.
As long as you’re not holding a gun to their head or stealing their identity and voting in their place, you can ask them to vote however you want. In the end, it’s the voter making the decision, regardless of their motivation. That’s pretty much how voting works. Romney, Gingrich and all those other guys are telling people how they think they should vote, and you have the same freedom of speech as they do.
Remember the general election of 2000?
Are you only asking British voters?
Probably asking the Canadian ones as well.
Many an election has been won or lost by a single vote.
I see no reason NOT to ask someone to vote and to suggest how they should vote. Billions of dollars are spent on ads doing exactly the same thing, but by political interest groups, parties and politicians.
Some people have no strong beliefs one way or another, so if you can sway them to go and vote, you might actually make a difference.
As an example, if I were a 17 year old Gay boy in California, you can damn well bet that I would be asking people go vote for equal rights propositions.
I would ask them instead of telling them; you might get more favorable results.
Depends on how much force you use or the amount you pay them. 
If you are extolling the virtues of a particular candidate and asking “can I count on your vote?” then you aren’t doing anything new. I don’t think the young Republicans (or Democrats) require that you be able to vote, do they?
If it helps, I asked my mom to vote for Obama but she could never vote non-Republican. Even my own mother wouldn’t do me that favor.
In very small electorates, maybe. Of the 16,000+ Congressional elections held in the US since 1910, only one was decided by a single vote. For larger elections for the Senate or Prez, I don’t believe its ever happened. And as a practical matter, a vote for a major office that came that close would almost certainly be decided by the courts.