suffrage (n.)

late 14c., "prayer," especially "intercessory prayers or pleas on behalf of another," from Old French sofrage "plea, intercession" (13c.) and directly from Medieval Latin suffragium, from Latin suffragium "support, ballot, vote cast in an assembly; right of voting; a voting tablet," from suffragari "express public support, vote for someone."

This is of uncertain origin. It is conjectured to be a compound of sub in some sense, perhaps "up from under" (see sub-) + fragor "crash, din, shouts (as of approval)" or its relative frangere "to break" (from PIE root *bhreg- "to break"). One theory (Watkins, etc.) is that the notion is "use a broken piece of tile as a ballot" (as in ostracism). But de Vaan writes:

Clearly, these forms are based on a verb *sub-fragere or a noun *sub-frag-o 'support'. Since the oldest meanings all refer to the process of voting for or against a candidate, one might support the view that *sub-frag- belongs to frangere 'to break':  *sub-frag- 'who shout in support of a candidate' (but fragor 'noise' is not attested before Lucr[etius].).

He also cites another modern theory that derives the -fragium from a root meaning "hind part, tail-bone" and "hypothesizes for suffragium a semantic change from 'support under the hind part' to 'political support'," but this he finds "too fantastic to be credible."

The meaning "a vote for or against anything" is from 1530s; specifically "a vote or voice in deciding a question or in a contest for office" by 1590s. By 1660s as "act of voting in a representative government;" the meaning "political right to vote as a member of a body" is first found in the U.S. Constitution, 1787, in reference to the states. Also sometimes "the collective opinion of a body of persons" (1570s).