My job is to photograph mushroom spores at x100 using immersion oil.
What is the best practice for cleaning the oil from the lens?
Right now I am just wiping it with a cotton swab.
In my experience, any microscope worth using has a minder. I would strongly suggest finding that minder, their opinions about cleaning practice should be respected. Objective lenses can be very pricey so they should be handled with care.
Detergent or water are inappropriate for an oil immersion lens. Detergent is likely to damage anti-reflective coatings on the lens. Water doesn't mix with oil.
My standard procedure with oil objective lenses is to wipe oil off with a good quality lens tissue (Kodak). Do not use facial tissues or a lab tissues, such as kimwipes. They are too abrasive and may sractch the lens. If someone has allowed oil to dry on the lens, I take care to know what solvents are correct for the lens and oil that are being used.
The least possible amount of cleaning naphtha, pharmaceutical quality (in German: "wundbenzin") and high grade lens cleaning paper. In the ol' days microscopists used some saliva on a piece of cloth, but that was in the pre-coated optics era.
Clean immersion oil from a lens by first pulling a fresh piece of lens paper slowly across the lens, like a bath towel across the back, to absorb the oil; shift to a fresh area of the sheet as required, and use more than one sheet if necessary (although that suggests you my have used too much oil). Then gently wipe the lens with fresh lens paper with a bit of solvent on it; for the most common Cargille type A and B oils, use a high grade of naptha or xylene (xylol). Finish by wiping with dry paper.
If a residue persists, and it will at least part of the time, clean the lens with lens tissue after gently breathing on the lens to fog it up: you want condensed moisture, not saliva. So it is a gentle, long h-a-a-a-a. Wipe in concentric circling motion, drifting in or out. Repeat. This method assures you will not wet and possibly harm the lens. Dried up oil can be softened with a trace of new oil, or better yet, one of these solvents. So yes, Always Confused, you still are; you need to remove that final trace of xylol. Don't use detergent. Or water. Ever. Use Xylol or naptha!
The range of answers here shows the dangers of soliciting online advice. Look at the online microscope "universities" put up by Zeiss and others, specifically, if possible by whomever made YOUR microscope. That's where to get the right answer. Today,even"minders" may not know!