claim (v.) Look up claim at Dictionary.com
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 PIE *kele- (2) "to shout," imitative (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."
claim (n.) Look up claim at Dictionary.com
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.