Can Botox Help Increase Facial Skin Thickness?

Botox will NOT increase skin thickness. The muscle "relaxation" of Botox may, over time and with regular treatments, actually soften the bulkiness of the facial muscles. Just as a body builder who quits lifting will lose muscle mass, regular use of Botox will decrease the size of the "overbuilt" muscles of facial expression, or any muscle it is used upon.

No, Botox use does not result in changes to skin thickness. The skin does thin naturally as we age, and that is a natural process that is unrelated to Botox use.
Botox can be used to purposely reduce muscle size. For example, patients who have masseter hypertrophy (a non threatening soft enlargement of the jaw muscles near the angle of the lower jaw), regular Botox treatments can result in weakening and atrophy of those muscles, creating a smaller appearance.

Botox blocks the transmissions of signals from the nerve to the muscles under the skin. It does not alter the dermal thickness directly. If one has a high resting activity of the contractile muscles before botox then the thickness of the skin and muscle per unit area might be more because of the corrugation of the muscle. After the botox the muscle doesn't contract so it may spread out to its flatter state making the thickness less. If you measure that thickness it would be back to normal.