claim (n.) Look up claim at
early 14c., "a demand of a right; right of claiming," from Old French claime "claim, complaint," from clamer (see claim (v.)). Meaning "thing claimed or demanded" is from 1792; specifically "piece of land allotted and taken" (chiefly U.S. and Australia, in reference to mining) is from 1851. Insurance sense is from 1878.
claim (v.) Look up claim at
c. 1300, "to call, call out; to ask or demand by virtue of right or authority," from accented stem of Old French clamer "to call, name, describe; claim; complain; declare," from Latin clamare "to cry out, shout, proclaim," from imitative PIE root *kele- (2) "to shout" (compare Sanskrit usakala "cock," literally "dawn-calling;" Latin calare "to announce solemnly, call out;" Middle Irish cailech "cock;" Greek kalein "to call," kelados "noise," kledon "report, fame;" Old High German halan "to call;" Old English hlowan "to low, make a noise like a cow;" Lithuanian kalba "language"). Related: Claimed; claiming.

Meaning "to maintain as true" is from 1864; specific sense "to make a claim" (on an insurance company) is from 1897. Claim properly should not stray too far from its true meaning of "to demand recognition of a right."