It’s hardly etymology. It’s morphology you fight with.

“Подождить” is a rare word meaning “to rain for some time.”

“Подождать” is the perfective form of “ждать,” which we can divide into the prefix “подо-” (being just a variant of “под-” modifying the sense of the stem into “littleness” of action) and the stem “ждать” (“wait”). So, it means “to wait a little time.”

The imperfective would be “поджидать.”