Me, I naturally call them by their original and proper name: les îles Malouines. The 'islands of the Malouins'.

Malouins are the inhabitants of Saint Malo, a maritime part of Brittany (Bretagne, not Great Britain). They settled and named the islands after them, including the capital of Saint Louis. The Spaniards, who briefly took over possession after the French, pronounce Malouin as Malvine. Hence they bastardised 'îles Malouines' into 'Islas Malvinas'.

îles Malvinas or îles Falklands are in common usage in French too.