Saul's name is derived from שאל, meaning "ask, request," and probably means "requested from God" (as noted by Daat Mikra; refer to 1 Samuel 1:20).
Sheol, meaning "grave, underworld," is of unknown etymology, and does not appear to have a regular Hebrew root. It may be an Assyrian loan word, but this is disputed.
None of the ancient commentators say anything about the similarity of these words. However, R' S.R. Hirsch, a 19th-century German rabbi, compiled a Hebrew dictionary which includes such word-similarities. He writes that Sheol (grave) is from שאל (request), for "the grave requests the return of the body" (see Genesis 3:19).
According to the (Christian) "Gesenius Hebrew Lexicon," the root אול means "foolish," and is used in the form אויל. Hirsch translates this root as "vacilate, lack clarity of purpose." Neither dictionary nor any commentators (that I've found) connect this root to either Saul or Sheol.