Preheat oven to 325 degrees F. Before starting batter, wash 8 (1-pint) wide mouth canning jars with lids in hot, soapy water and let them drain, dry, and cool to room temperature. Generously grease inside of jars.
Drain crushed pineapple for about 10 minutes in a colander, reserving juice. Puree drained pineapple in a food processor. Measure out 1 1/2 cups puree, adding a little juice if necessary to make 1 1/2 cups. Set puree aside. Discard remaining juice or reserve for another use.
With an electric mixer, beat together butter and half of brown sugar until light and fluffy. Beat in eggs, then remaining sugar. Beat in pineapple puree and rum and set aside.
Sift together flour, baking powder, and baking soda. Gradually add to pineapple mixture in thirds, beating well after each addition to make a thick batter. Stir in coconut.
Spoon 1 level cupful of batter into each jar. Carefully wipe rims clean, then place jars in center of preheated oven. Bake 40 minutes.
About 10 minutes before cakes are done, bring a medium saucepan of water to a boil. Put in jar lids, cover, and remove from heat. Keep lids in hot water until they're used.
When cakes are done, remove jars from oven. If jar rims need cleaning, use a moistened paper towel. Carefully put lids and rings in place, then screw tops tightly shut. Place jars on a wire rack; they will seal as they cool.