Old English pytt (Kentish *pet), "natural or man-made depression in the ground, water hole, well; grave," from Proto-Germanic *putt- "pool, puddle" (source also of Old Frisian pet, Old Saxon putti, Old Norse pyttr, Middle Dutch putte, Dutch put, Old High German pfuzza, German Pfütze "pool, puddle"), an early borrowing from Latin puteus "well, pit, shaft."
The Latin word is perhaps from PIE root *pau- (2) "to cut, strike, stamp," but there are phonetic and sense objections.
Short u makes it impossible to directly derive puteus from paviō 'to strike'. It might be related to putāre 'to prune', but this is semantically less attractive, and the suffix -eus can then hardly be interpreted as indicating a material. Therefore, puteus may well be a loanword. [de Vaan]
Meaning "abode of evil spirits, hell" is attested from late 12c. Meaning "very small depression or dent in the surface of an object" is from early 15c. The anatomical sense of "natural depression or hollow in some part of the body" is by late 13c,; the pit of the stomach (1650s) is so called from the slight depression there between the ribs; earlier words for it were breast-pit (late 14c.), heart-pit (c. 1300).
The meaning "part of a theater on the floor of the house, lower than the stage," is from 1640s; the sense of "that part of the floor of an exchange where business is carried on" is by 1903, American English. The pit dug under a large engine or other piece of machinery to allow workers to examine or repair it is attested by 1839; this later was extended in auto racing to "area at the side of a track where cars are serviced and repaired" (by 1912).
mid-15c., "to put or set in or into a pit," from pit (n.1); especially for purposes of fighting (of cocks, dogs, pugilists) from 18c. Hence the figurative sense of "to set in rivalry, match as opponents" (1754). Compare pit-bull as a dog breed, attested from 1922, short for pit-bull terrier (by 1912). The meaning "make pits in, form a little pit or hollow in" is from late 15c. Related: Pitted; pitting.