This might look like one of the many words ending in -ee for a person affected in some manner by the action of the associated verb, such as employee, inductee, internee, interviewee, or licensee. But the ending is accidental, as in dungaree, squeegee and jamboree.
A raparee was once an Irish foot soldier, armed with a weapon that in Irish was called a rápaire. This is a cousin of English rapier, though not the same sort of weapon, the Irish one being a short pike. The Irishmen so armed were an irregular soldiery who fought on the Catholic Jacobite side during the war that William III waged in Ireland in 1689-91. During and after the war, many took up thieving and banditry, which is why their name is thought also to contain another Irish word, rapaire or ropaire, a violent person, irregular soldier or robber.
Raparees became a menace and in 1695 the government passed An Act for the Better Suppressing Tories, Robbers, and Rapparees. Tory, the nickname of members of the Conservative Party in the UK, is from Irish toraidhe, a highwayman or outlaw, and initially referred to Irish peasants dispossessed by English settlers and living as robbers. It was taken up as a term of political abuse in the 1680s for those who opposed excluding the Catholic James from succeeding to the English crown.
Raparee is now solely of historical interest, but this modern example demonstrates that its relevance soon spread far beyond Ireland:
We were building a schooner from the wreckage when a horde of ill-favoured raparees attacked us — Dyaks and Malays led by a nasty confident quean, a bloody-minded covetous froward strumpet.
The Nutmeg of Consolation, by Patrick O’Brian, 1991. A quean was an impudent or badly behaved woman, from Old English cwene, a woman, hence also queen. A froward person was one difficult to deal with.
The idiom not giving a rap, meaning not caring, is said to be connected. In the eighteenth century, Ireland was short of coinage and counterfeit equivalents of coins of small value such as the halfpenny or farthing were widely used instead. These came to be known as raps, in part from Irish rap for a bit or piece but also as a link to raparee. The idiom is recorded from later in the century.