The principle goals of the Stack Exchange network are to to help disseminate information and settle disputes through productive peer review whenever it is possible to do so, and help future researchers more easily find the answers to whatever questions they may have. It does not really matter how much reputation someone has, just so long as their post helps us to succeed in these goals. The reputation system is mainly meant to help us achieve those goals, rather than to serve as a detriment to them.
As a result, when deciding whether or not to vote for a question's closure, I would advise ignoring the amount of reputation a person has and applying the standards just the same. I think the minimal threshold of necessary research varies on a question by question basis, and should be low enough to accommodate newer users while remaining just the same for even the more experienced ones as a matter of fairness.
However, that is just my opinion, and we might not be able to get everybody to agree on that, so it would be advisable for you to do as much as possible anyway. Remember that questions which can be skillfully crafted to generate intrigue will be rewarded with more votes for them than they would be otherwise.
As such generally speaking I would suggest going ahead and researching the question as if you were attempting to write an answer. Document what research efforts failed, and which ones succeeded. Consider if it was too easy to find the answer to the question using methods you would expect a lower reputation user to use in a valid question, or if it is legitimately obscure information before deciding to post it. Defer to the following flowchart from Are Some Questions Too Simple? when deciding if the question should be asked on our website, while keeping in mind that if it should not be answered, it should not be asked either:
Note that fact that this flowchart gives a question every opportunity to be salvaged if it has any redeeming value. Pay special attention to the last point. It is there for a reason. Maybe there are special circumstances that render the answer to the question especially interesting, such as the conventional wisdom about the matter being wrong for instance. Just keep in mind that the community reserves the right to make the final decision regarding what it considers to be "interesting", either with or without basis, with its votes though. Preferably, the test should succeed before it ever gets far enough for this to be a consideration.
Also, use your experience with our website to avoid asking questions that would be closed for other reasons. Don't Ask primarily opinion based questions, or true duplicates. You should also write questions clearly, and explain how any potential duplicates that might be found differ from yours. You know this already though.
If after referencing these guidelines you decide that the question can be productively answered, then disclose as many of your failed research attempts as possible in the body of the question. Include as many as you can reasonably fit into the body of the question in order to prove to others that it is not too easy for people to find the answer otherwise, and prevent others from making unnecessarily redundant research efforts.
If you do find a good answer to such a question, please do not keep it to yourself: Publish the answer alongside its question, as well as you can possibly document it. We want to learn the answers to difficult or interesting questions, or at least help future visitors who may have the same problem to find them. There is a checkbox which enables this function to expressly encourage this behavior on every Stack Exchange website:
You have all of the time in the world to document why you think this is the best answer, but it is actually possible that answer you found is not the best one, and it should be submitted for peer review. Simultaneous publishing is expressly provided for and implicitly encouraged with a checkbox that can be found beneath the "post your question" button, which gives you a separate field in which you can start writing an answer.