As Don said, you can use baby oil which is basically scented mineral oil. If you have to go out and buy some (grocery store/Walmart), might as well get the mineral oil though since it is unscented.

Process is pretty simple. Take either a cotton ball/cloth/unused foam dabber, get a tiny amount of mineral oil on it.
You really don't need much. Gently pass it along the outside edge of the chip, then one face of the chip. Leave overnight to dry. Next day flip over and do other face of chip, leave to dry all day.
Some people do both faces at the same time, I have never tried it.

At first the chip will look TOO oily and the colors may pop too much. Once dry the colors will look great though.
I only do this to less valuable chips that I actually play poker with.

In the two pictures here you'll see some $.50 Fitzgerald's. One shows a chip just oiled vs non-oiled.
The other picture with stacks of chips is a comparison the next day after drying. The 1st chip in the top row was oiled, the other 3 chips are dry and dull looking.

It's now 6+ months after oiling the chips, you can still see the difference.
(I just noticed I forgot to oil one row of chips so was able to compare.)