The generally accepted practice, as I've read on numerous websites and through books, is to break up the dried up clay into small pieces, add a little mineral oil (or baby oil), and massage/work the oil in. (If you're not wearing gloves, this will 'stain' your hands but it will wash off with a little effort). After the smaller pieces are softened, merge them back together and continue to work the larger lump to spread out the oil evenly, as you normally would to soften the clay.
Sculpey also markets a clay softener that helps with this process, however if too much is used it can affect the color of the clay, or make it slightly translucent.
When adding oil it is possible to add too much, and this will be evident in the clay. To remove excess oil, wrap the clay in several layers of an absorbent material, such as paper towels, and put it in a ziplock bag. Leave to rest for a day or two, checking the consistency occasionally and replacing the absorbent material, until you have the consistency you desire. This process will tend to wick the oil out of the outer portion of the lump of clay more than the inner portion, so after removing the absorbent materials and massaging the clay again, it will be slightly softer than indicated before massaging (how much will depend on the size of the lump prior to wrapping it.)
Adding mineral oil to soften clay also makes it easier to blend different colors to make a custom color, and then remove the excess oil using the process described above.