"upper limb of the human body," Old English earm, from Proto-Germanic *armaz (source also of Old Saxon, Danish, Swedish, Middle Dutch, German arm, Old Norse armr, Old Frisian erm), from PIE root *ar- "to fit together" (source also of Sanskrit irmah "arm," Greek arthron "a joint," Latin armus "shoulder"). Arm of the sea was in Old English. Arm-twister "powerful persuader" is from 1915. Arm-wrestling is from 1899.
They wenten arme in arme yfere Into the gardyn [Chaucer]
"weapon," c. 1300, armes (plural) "weapons of a warrior," from Old French armes (plural), "arms, weapons; war, warfare" (11c.), from Latin arma "weapons" (including armor), literally "tools, implements (of war)," from PIE *ar(ə)mo-, suffixed form of root *ar- "to fit together." The notion seems to be "that which is fitted together."
Meaning "branch of military service" is from 1798, hence "branch of any organization" (by 1952). Meaning "heraldic insignia" (in coat of arms, etc.) is early 14c., from Old French; originally they were borne on shields of fully armed knights or barons. To be up in arms figuratively is from 1704; to bear arms "do military service" is by 1640s.
Old English side "flanks of a person, the long part or aspect of anything," from Proto-Germanic *sīdō (source also of Old Saxon sida, Old Norse siða, Danish side, Swedish sida, Middle Dutch side, Dutch zidje, Old High German sita, German Seite), from adjective *sithas "long" (source of Old English sid "long, broad, spacious," Old Norse siðr "long, hanging down"), from PIE root *se- "long, late" (see soiree).
Original sense preserved in countryside. Figurative sense of "position or attitude of a person or set of persons in relation to another" (as in choosing sides) first recorded mid-13c. Meaning "one of the parties in a transaction" is from late 14c.; sense in a sporting contest or game is from 1690s. Meaning "music on one side of a phonograph record" is first attested 1936. Phrase side by side "close together and abreast" is recorded from c. 1200. Side-splitting "affecting with compulsive laughter" is attested by 1825.