Wordorigins.org

This phrase meaning ill or indisposed is an Americanism dating to 1827. From the Austin Papers of that year in a reference to the unsuccessful Fredonian Rebellion of 1826-27, when a small group of Texans attempted to oppose the Mexican government:

The fredonians is all here rather under the wether.

The phrase probably derives from the idea that the weather can affect your mood and health. It is often thought to be nautical in origin, but there is no evidence of that.

(Source: Oxford English Dictionary, 2nd Edition)