If one wanted more basement ceiling height
The safer and more predictable method (probably also cheaper) is to jack the house and build up, not messing with the footings. If the lower foundation is in good shape this can be a relatively simple approach if you work in concrete masonry unit lifts (i.e. an evenly divisible by 8 inches lift for most concrete blocks - perhaps 3 blocks or 2 feet for yours.) You raise the house somewhat more (well, unless you are nuts, your professional house movers do) and build the new block wall (reinforcing  probably a good idea to ask your engineer about) on top of the old wall, then set the house back down on it. 

Another fairly common option with more dire foundation issues is to either build a whole new foundation and roll the house onto it, or roll the house off to the side and build a new foundation in place, then roll the house back onto it. This requires space to do that, but it is considerably cheaper to move the house than to build the foundation while the house is in the way. However, the latter approach is also seen.