The community of all users, high and low reputation alike, already has a method for marking questions as proposed solutions in the form of upvotes. Pretty much by definition, a useful answer is one that can solve the problem. Yes, this does have problems with joke answers that get upvotes, like a certain ASCII ghost, but past that the general process is that the community upvotes indicate what the rest of the community thinks is a viable solution.
Ultimately, the "acceptance" marker indicates that a solution was successful for the author. The community does not have a voice in this, and never should. It doesn't matter how many other people were able to adapt a particular solution, if it doesn't work for the author then it should not be marked. The accept checkmark even indicates in the tooltip that it is what the author found to work. The lack of an acceptance mark may simply mean that the author is still deliberating on which works best.
This is why you have badges like Populist, where another answer gets far more votes than the accepted answer. This is the indication of what may be a more generally usable solution, without speaking (perhaps falsely) in the name of the original author.
There is also no way of knowing whether the author is still around or not. Nor should the author have to override the decision of the community as to what answer the author found most useful.
Finally, with regards to reputation limits, that is not very useful to implement. Reputation of a user has no impact on their knowledge of whether an answer is good or not. A low reputation user who knows the situation and may have encountered the problem in the past should not be less qualified than a high reputation user who has never seen the stuff before. That user should not be more qualified, either, because outside of the author's action we do not specifically value any one user's opinion greater than any other. So votes remain a very equalizing measure that's basically available to everyone.