Boil rum and raisins in small saucepan 30 seconds then remove from heat and let stand while making pudding.
Bring water with salt to a boil in a 3 to 4 quart heavy saucepan. Stir in rice and return to a boil. Cook, covered, over low heat until water is absorbed, about 18 minutes.
Stir in milk and sugar and simmer, uncovered, stirring often, until thickened and rice is tender, 40 minutes to 1 hour. Stir in vanilla, rum soaked raisins. Cool as desired. Pudding can be eaten warm or chilled.
Pudding can be kept chilled 5 days.