Old English þuma, from Proto-Germanic *thūman- (source also of Old Frisian thuma, Old Saxon, Old High German thumo, German Daumen, Dutch duim "thumb," Old Norse þumall "thumb of a glove"), literally "the stout or thick (finger)," from PIE *tum- "swell," from root *teue- "to swell." Unetymological spelling with -b (attested from late 13c.) is perhaps by influence of dumb; also compare limb (n.1).
In some of the IE languages there is a single word for "thumb," which is called the "big finger," like NE big toe. Many of the single words are of similar semantic origin, based on the notion of "stout, thick." [Buck]
Compare Greek megas daktylos "thumb," but Greek also had antikheir, literally "what is opposite the fingers." Italian pollice, French pouce are from Latin pollex, perhaps formed (on analogy of index) from pollere "to be strong."
Phrase rule of thumb attested by 1680s (the thumb as a rough measure of an inch is attested from c. 1500). To be under (someone's) thumb "be totally controlled by that person" is recorded from 1580s. Thumbs up (1887) and thumbs down (1906) were said to be from expressions of approval or the opposite in ancient amphitheaters, especially gladiator shows, where the gesture decided whether a defeated combatant was spared or slain. But the Roman gesture was merely one of hiding the thumb in the hand or extending it. Perhaps the modern gesture is from the usual coachmen's way of greeting while the hands are occupied with the reins.