The meaning of “to swing the election”

In the US, run for office is much more common than either run for election or stand for election. Consider this NGram (American English corpus):

Thus, if you are in the US, I'd agree with the correction you were suggested, especially since run for [office] can be and usually is used with the name of a specific office. Since your sentence talks about not just a generic election, but election to a lodge office, it seems a little redundant to say standing for election to a lodge office when you could just say running for a lodge office.

Addendum: Though this wouldn't really fit in the example sentence you gave, if you want a stock phrase of similar meaning that uses the word election, Americans often say that a person is up for election. e.g. "Bush was up for election in 2000, and again in 2004."

This revolves around a couple of related idioms, actually, related to baseball and softball.

Let's say two people are engaged in a debate. One says something that makes it easy for the other to make an incontrovertible point, and win that round of the debate. In the debating circles, the act of making a memorable, incontrovertible point is sometimes called "hitting a home run," and the gaffe that makes it easy for that home run to be hit is sometimes called "an easy pitch" (or "a softball"). One renowned softball I remember was when Dan Quayle spoke of how his relatively young age shouldn't be in issue in him being qualified to serve as president, if needed. His opponent, Lloyd Bentsen, was ready for that pitch.

Sometimes this baseball/softball analogy is applied to interview situations as well. If a journalist giving a recorded interview doesn't ask any hard questions ("hard" in this case means something that would make the interviewee uncomfortable, such as questions about a recent scandal, perhaps), that is sometimes described as softball questions, or a softball interview (see this headline, for example: Kroft's Softball Obama Interviews Diminish '60 Minutes'). In a similar way, an interviewer who asks an unexpectedly difficult question might be said to have "thrown a curve ball," or one who asks a lot of tough questions relentlessly might be described as "playing hardball" during the interview.