In a 4- to 6-quart heavy pot crush 1 cup of the blackberries with a potato masher. Continue adding berries and crushing until you have 6 cups crushed berries. Stir in pectin and sugar. Bring mixture to a full rolling boil, stirring constantly. Stir in honey. Return to a full rolling boil, stirring constantly. Boil hard for 1 minute, stirring constantly. Remove from heat and quickly skim off foam with a metal spoon. Stir in thyme.
Ladle hot jam into hot sterilized half-pint canning jars, leaving a 1/4-inch headspace. Wipe jar rims; adjust lids and screw bands.
Process filled jars in a boiling-water canner for 10 minutes (start timing when water returns to boiling). Remove jars from canner; cool on wire racks. Makes 8 half-pints.