This is a real problem for me. I keep my ways oiled, and chips fall on them and stick to the oil.
Also, I apply oil to the workpiece and to the tool tip, which I have read is important, because it cools and lubricates the cut.
Not being obnoxious, I'm genuinely interested to know how you keep oil away from your chips so I can shamelessly copy you
My version of the lathe has oil ports and way wipers in the carriage, and I only apply a tiny bit of oil, so there's the thinnest of films on the ways.
My understanding is that the greatest benefit of using cutting fluid (which may or may not be oil) is to keep the temperature of the tool from rising to the point where it affects the temper. This, I think, is mostly applicable to HSS (high speed steel) tooling, and the need for this measure is related to the size of the workpiece, the rotational speed of the spindle, and the depth of the cut, as well as the material being turned (and probably other factors I'm forgetting). I don't think I have ever had a combination of those factors that came anywhere close to causing a cutting tool to hit a temperature that would have affected its temper; these little lathes just aren't big or muscular enough to get to that zone. Lubrication can also produce a better finish.
When I do feel like a bit of lubrication is needed to make a cut easier or better, I like to use a Tap-Ease stick (very convenient and not messy), or WD-40 (which is essentially kerosene) for aluminum. I do use some cutting lube when drilling, as there is a lot of friction involved, and it helps quite a bit; WD-40 is great for this situation, as you can squeeze off a little shot right into the hole. WD-40 also seems to go away pretty fast, and it has the side benefit of displacing water (WD...get it?), so it can help keep things from rusting.
However, I very rarely use any coolant or lubrication when performing turning operations on the minilathe. I just don't see the benefit -- plus, as you've found, it can make a mess.