Some claim it is a variant of the Greek word Phoebos, meaning "bright," which is why it goes together with Shraga (= flame in Aramaic).
However, others dispute this etymology. Here is one interesting alternative:
"Feivel" does not come from the "Phobos" (aka Apollo), the Greek Sun God! (This was a false etymology put about by German Jewish scholars in the 19th century.) Etymologically, the earliest instances of the the name "Feibush" are written as "Vivs" (Vivus) which is the Latin equivalent of "[C]Haim", "life", a name which was given to sick children (and adults) to "confuse the Angel of Death" (who could then no longer find the person named on his "list" who he had to kill).
Stated a bit differently here (scroll about halfway down):
Uri, Shraga and Phoebus (of which Feivish and Feivel are yiddishized corruptions) all mean basically 'bright', 'shining', 'light'.
"Fayvush or Faivel are the kinnuim of Shraga. Fayvush is an ancient Jewish name whose origin was the Latin vivus (living, alive), a loan translation (calque) from Hebrew chaim (life). Later the name Fayvush was erroneously considered to be a derivation from Phoebus, god of the sun--consequently Fayvish became the kinnui not only for the Biblical Hebrew name Uri (light) but for the Aramaic name Shraga (candle) in the Rabbinic period. Folk legends along with true and false etymologies gave rise to these interwoven associations between Hebrew name and kinnui.