In a medium saucepan, combine cranberries, sugar, water, 3 Tbsp rum, lime juice, 1 tsp lime zest, and 1 Tbsp chopped mint. Stir to combine.
Over medium heat, bring mixture to a boil and then lower heat and let simmer for about 10 minutes minutes, until cranberries start bursting.
Remove saucepan from heat and stir in remaining 2 tsp rum, 1 tsp lime zest, and 1 tsp chopped mint.
Let cranberry sauce cool a bit before placing in fridge to cool completely. It will thicken up as it chills.