claim

1250–1300; (v.) Middle English claimen < Anglo-French, Old French claimer < Latin clāmāre to cry out; (noun) Middle English < Anglo-French, Old French cla(i)me, noun derivative of the v.

1. See demand.

The English noun claim comes from the verb, which in turn comes from the Old French verb clamer (stem claim- ) “to summon to law, affirm vigorously,” from the Latin verb clāmāre “to shout, accompany with shouts, shout the name of.”
The verb claim originally meant “to assert a legal right, to make a demand for something that is one’s due.” In the 19th century, claim developed a looser, less strict sense, especially in American usage, “to make an unsubstantiated statement; assert or maintain as a fact,” a meaning considered inelegant at that time but also one that occurs in the writings of Chaucer.
The legal term quitclaim meaning “to quit or give up a right or claim” dates from the 14th century in England. The noun claim meaning “a request or demand for payment in accordance with an insurance policy” dates from the 19th century.
Just as we intuitively understand the relationship between claim and quitclaim, it's pretty clear how the words acclaim, reclaim, proclaim, etc., are related in meaning and etymology to claim.