My mother lives in a mobile home park in Villa Ridge, Missouri and the day before Thanksgiving she got a notice that they are increasing her pad rent 150% beginning in January. That seems like there should be something that prevents them from more than doubling her rent. Is there anything to protect her?

Yes there seems like there should be but I doubt there is. Normally rent is based on what the market can hold, If for example the current going pad rent is that high, then they can ask for it.

How does this new price compare with other lot rental in that area?

In some places they do have rent controls, but I don't think trailer pads are part of it, and not in your area either. You can and maybe should contact a real estate attorney and see if there is anything you can do.