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 root *kele- (2) "to shout." Related: Claimed; claiming.
Meaning "to maintain as true, assert a belief or opinion" is from 1864 ("A common use, regarded by many as inelegant" - Century Dictionary, 1895); claim properly should not stray too far from its true meaning of "to demand recognition of a right." Specific sense "to make a claim" (on an insurance company) is from 1897.
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); claim-jumper is attested from 1839. Insurance sense "application for guaranteed compensation" is from 1878.