Foundation raising or a new foundation
Most houses that have moved don't need a new foundation, because the slab or the framing is usually sound and only needs firm support under it again, so lifting onto new piers is what I recommend in most cases. What I won't do is patch the cracks and stop there. It's the first place people try to save money, but the cracks tend to come back while the ground keeps moving, and the support underneath is where the money belongs.
What shapes the cost of foundation raising
A house on a crawl space usually costs less to level than a slab, because we can reach the framing from underneath instead of digging along the walls. On a slab, the price follows the pier count, and even one settled corner takes several piers. How deep the firm soil sits and how much room we have to work decide between push and helical piers, and stucco or brick veneer adds work on a whole-home lift because it has to be removed or braced to stay in place.