1. Rinse a small teapot with hot water, then drain. Add mint leaves, brown sugar, tea bag, and 1 cup boiling water; cover and let stand for 2 minutes, jiggling tea bag several times. Lift tea bag out and add rum and lime juice.
2. Pour mojito tea through a fine strainer into warmed heatproof martini or wineglasses (at least 1-cup size; see notes).