Solution 1:

Compared to some other resources (like quarried stone or wood) not all that much.

The Elder Scrolls Wiki article on Homesteads has handy tables for determining the total costs of every stage of building (to view tables that include furnishings, you may have to delve slightly deeper and click on the links to individual build items). Each Homestead has the same building stages: Small house > Main hall > Add 3 Wings. Ultimately, unless you build each house with the exact same wings, costs will vary between them.

However, since you are interested in knowing the maximum amount of clay you will ultimately need to mine, here are the clay totals including furnishings (I've included the clay costs of the remaining 6 wings for comparison):

  • Small House: 4
  • Main Hall: 24
  • North wing, Trophy Room: 13
    • Storage Room: 3
    • Alchemy Laboratory: 9
  • East wing, Library: 9
    • Armory: 3
    • Kitchen: 5
  • West wing, Greenhouse: 21
    • Bedrooms: 3
    • Enchanter's Tower: 9
  • Exterior: 6

For a maximum total of 77 clay per homestead, or 231 clay for all three.