I am fed up with experienced users who get all bent out of shape about whether askers accept their answers. You have plenty of reputation to participate fully on the site, so this is clearly just an ego thing.
Perhaps the asker didn't find your answer helpful or it didn't actually solve their problem. Why should they accept your answer in that case?
Beyond that, a variant of your proposed solution already exists. When users have less than 1k reputation, they get a pop-up reminder to accept the answer whenever they upvote it. It says:
don't forget you can mark this as the accepted answer by clicking its check mark
Not to mention there's a huge, hollow check mark glyph in the margin of the answer. Since people are naturally curious (and since they've seen this arrow filled in on other people's questions), they might wonder what in the world this icon means or does. If they hover over it, they'll see a tooltip that says:
Click to set this answer as your accepted answer; click again to toggle
People who ignore the existing UI prompts aren't likely to pay any attention to obnoxious animation, either. That will just annoy people who do follow instructions, the kind of users we'd like to keep around.
If you honestly think that your answer is helpful to a new user, but they haven't read the FAQ and don't understand how to accept an answer, you could leave a helpful comment in the following vein:
Welcome to Stack Overflow! If my answer helped to solve your problem, please consider marking it as accepted. That's the customary way of indicating that your question is "resolved" and thanking the person who helped you.
But be nice, otherwise people will flag these comments as rude/offense, and they will get removed.