Rinse the berries and discard any that are bruised. Place the berries in a large pot with a heavy bottom.
Submerge the peaches in boiling water for 30 seconds. Remove them from the water and submerge them in cold water. Cut each peach in half, and then it should be very easy to remove the skin. Chop the peaches and place them in the pot with the berries.
Add the lemon juice and 1/2 cup of honey.
Cook over medium heat, partially covered, for 20 minutes. The mixture will become very watery early on, and it will thicken as you cook it.
After the first 20 minutes, mash the fruit with a potato masher.
Simmer uncovered for another 30-40 minutes, stirring regularly. Taste the jam and see if it seems too tart. If it does, add another 1/4 cup of honey. If it still seems too tart, add another 1/4 cup of honey for a total of one cup of honey. (I would have stopped at 3/4 cup honey if it were just me eating it, but I knew my family would appreciate that last 1/4 cup.)
After a total of 50-60 minutes on the stove, you will notice that the jam looks nice and thick. You can test if it’s done by taking a teaspoonful of jam and dropping it back into the pot. If the jam runs right off the spoon, it’s not quite done. If it's a bit sluggish running off the spoon, it’s done.
To can this recipe, sterilize four half-pint jars, lids, bands, and a spoon by running them through the dish washer and timing them to be done when you’re ready for them. You'll probably only need three jars, but it's good to have an extra one ready just in case. The lids should be sterilized in hot but not boiling water (180 degrees) if you don’t use the dishwasher to sterilize them.
While the jam is cooking, boil a big pot of water with a canning rack at the bottom of the pot. The water needs to cover the jars by at least an inch.
With the sterilized spoon, scoop the jam into the hot jars without touching the inside of the jars with your finger. You don’t want to introduce any bacteria to the jam’s environment. A wide canning funnel facilitates this process.
Leave 1/4 inch head space at the top of the jars and put the lids on top of each jar. Don’t touch the bottom of the lid.
Loosely screw on the bands until they are just connected. You need to leave them loose so they process correctly.
Carefully place the jars in the boiling water bath for 10 minutes.
Turn off the heat after 10 minutes and leave the jars in the water for 5 more minutes.
Take the jars out and place them on the counter. Within seconds, you should hear a little “pop” coming from each jar. This is the sign that they have sealed properly.