I would think it's beneficial because future viewers can see that an answer is the correct one, but apart from that, is there any other benefit to the site? I began politely prodding people to accept my answers, but I feel I should know if there is any tangible benefit to the site as a result, e.g. SEO benefits, before continuing. I also figured I would wait a week or so before prodding again. Otherwise, it's probably just annoying people without any net gain on the part of the site.
An accepted answer means that the Original Poster (OP) got the answer he is looking for. The only benefit is knowing that the OP got what he was looking for.
In most cases the accepted answer is right, but there are quite a few cases where other answers maybe be more apt than the accepted answer. It also motivates / gives satisfaction to people who have taken effort to answer. It's an explicit sign / thank you from the OP to the person who has answered.
I suggest that you do not prod them much. Let 'em be. If they understand what this is all about, they will accept those answers that they feel are correct. If they don't understand, you want them touching as little as possible.
Not all questions can be answered, so there shouldn't an expectation that every question will have an accepted answer.
The perspective is incorrect. Who says your answer is correct? If anything, they should be prodded to select an answer that they feel is correct, if one is available. Pestering them to accept your answer is incorrect.
An accepted answer is worth 15 rep. An up-vote is worth 10. Up-votes outstrip accepted answers by a long shot. We shouldn't worry too much about having accepted answers.
Overall, they are good and are to be encouraged. However, I see little value in pestering people about something relatively trivial. People should be encouraged to accept answers, but no hounded to do it. There's already at least one discussion about this on the main meta somewhere.
Be kind as you do it, and feel good that you are making the Internet a better place.
I believe Dheer is exactly correct, with some additions.
Pestering folks to accept your answer is fine; it reenforces the norms of the site. Our goal is to make the internet a better place, and we have rules and culture to support that mission. If a new users doesn't want to play by those rules, they don't have to, but they won't feel completely welcomed either. (Getting pestered to accept an answer might not be fun). Having a consistent culture makes the community stronger and focused. The benefits of a focused community is getting consistently high volumes of traffic to the site.
Consider the interested players in the site:
- StackExchange corporate wants pageviews. The more visitor, the more ads they can sell for a higher price.
- The visitors want good answers on a site that is legible. Voting should bring those good answers to the top, and an answer also promotes it.
- Members of the site want reputation and a good community to "discuss" their passion.
These goals are very aligned in output. Reputation is the gamification of the site; the means to an end.
To achieve their goal, StackExchange needs content. Since most people won't be digital sharecroppers for free, the points system acts as motivation for great users (such as yourself) to provide content.
The users get to have good, generally trustworthy answers they can be quickly evaluated. The site is pleasant to read, loads fast, and well placed in search engines.
Those who want to stick around get to play games of reputation. The site of experts also provides personal growth in areas that really interest them.
All of that ends up as a community that can make the Internet a better place. So feel free to
pester folks ask once to accept the answer. That just happens to be how this community works.
When visiting an SE site I consider it inauspicious if there are lots of unanswered questions, and many are dated 2010 or older. It makes the site seem desolate, not up to date, not relevant. We don't want that.
There's a difference between a question that has answers, and a month has passed, versus the ancient questions. After six months or a year has passed, it can get annoying. It also looks messy. We had similar concerns on Meta Cross-Validated SE, about questions with no accepted answer. A comment might completely answer the question. Yet the question would remain open for ages, because there wasn't an answer posted.
Accepted answers indicate that the person who asked the question derived value from Money SE.
EDIT: I found what I was looking for originally, a question on Meta SuperUser about "prodding protocol"! It even has an answer from Jeff Atwood (although it wasn't the accepted answer ;o) There are lots of comments. Most reinforce the answers given here, including George's and MrChrister's answers.