To a large pitcher, add water, brown sugar, and lime juice. Stir until the brown sugar is dissolved. Add mint leaves and stir.
Chill for 2 hours before serving so that the mint will infuse flavor into the limeade. If a stronger mint flavor is preferred, chill overnight. The mint leaves will just need to be removed before serving because they'll turn a dark color.
Just before serving, add ice to the pitcher.
Optional: Brown Sugar Rimmed Serving Glasses/Mason Jars - An option is to rub a wedge of lime around the rim of a serving glass or mason jar. Then put some brown sugar in a plate or small bowl, turn the glass or mason jar upside down, and dip it in the sugar until the rim is completely coated with the brown sugar.
Serve while ice cold. Garnish the serving glasses with lime wedges and sprigs of mint.