Add the tomatoes to a blender or food processor. Pulse several times to crush the tomatoes.
Transfer the tomato puree to a medium skillet set over medium heat. Stir in the brown sugar, lemon zest and juice, and ginger. Cook, stirring frequently, until the mixture is thick and jammy – this will take about an hour. (To test, dip a cool metal spoon into the jam. It should slide of the spoon in a thick sheet – not in watery droplets.)
Ladle hot jam into the jars. Store in the refrigerator or freeze.
Alternately, wipe the rims of the sterilized jars, cover with the lids, and screw the bands on until they're just barely tight. Seal in a hot water bath for 20 minutes. Carefully remove the jars from the pot and let them rest on the counter for at least 6 hours. You should hear the jars make a "ping" sound when they seal. The homemade jam is delicious right away, but tastes even better after a few weeks.