I am writing a research paper in which I mention Moroccan explorer Estaban quite often. In the books I have, his name is spelled in different ways; the Wikipedia article linked above lists even more variants, though I never saw a single scholarly source use the name "Mustafa Azemmouri". The problem I get when researching is, every scholar uses different spelling/variations on the name, so I have to search the 5 or 6 different spellings for articles and books to come up in my searches in ProQuest. How do I pick which name/spelling/variation to use?
You cannot make the different spelling variants disappear. So if there is no "official" spelling, all you can do is to pick one of the variants and use it consistently throughout your work. You might choose the most abundant version, or a version used by the person in question, or the most modern version, or the version used in your group...
If this is a critical aspect in your work, you could also include a short discussion of the different spellings of the name and why you picked a specific version. This way, you can also add some context, so the person can be identified more accurately.
Actually, I'm surprised that you would use his slave name rather than his known actual name. I would recommend his real name, Mustafa Azemmouri, for use in you paper with a footnote (or similar) pointing out the other names he was known by. That will make searches possible. A footnote would also make it clear who is being referred to when you quote older sources that use the slave name.
I would treat the history of the (mis)use of his name as being a racist artifact that should be dropped in modern usage. Honor the man. Use his real name.