I've jacked a house up (to improve drainage), but never dug a basement, although I may very well in the future....so I have been doing a little reading.
You have a few points that are very important. First, how deep are your footings? You cannot dig deeper than the bottom of your footing withough jacking up the house, breaking up the existing foundation/footing, digging and then pouring new ones. Quite honestly, unless this is a house of great value in a great location, this is probably going to be far more trouble than it is worth.
If you are in a more northern clime and the footings are down below the depth you wish to dig to, the foundation walls are in good shape, your house is at the right grade (no need to jack house up), and you have enough room to work, it may be well worth digging down to your desired basement depth. It also may be worth it to advertise at a local high school, offering 16/17 year olds $100 for a 9 hour Saturday and free food to do some of the labor of digging.
A couple of other considerations. How deep does your main soil pipe go? If not deep enough, you may have to install a lift station if you plan on putting in any kind of basement plumbing. If you plan on finishing the basement, be sure to install a French Drain around the perimeter of the foundation wall to a sump pump in the low corner. Stairs will have to be lowered, furnace plenum and water heater plumbing both extended (if you have an existing basement) Finally, might want to consider putting down rigid insulation and then running in-slab hot-water radiant heating tubing. A great way to heat your new living space.