Bike was new around a year old. 8 of the back spokes broke and it went out of true. I had those repaired with a slightly stronger but spoke but now 2 more have broken including one of the thicker ones and it's out of true again. I have a kiddie seat on the back and am wondering if this is somehow contributing to the breakages. Is there anything I can do to get a very strong spoke? They are all breaking at the j hook location.
Usually spokes don’t break because they are too weak but because they have too low or uneven tension.
Even tension will distribute the load among several spokes. High enough tension will make sure that spokes are never unloaded completely since spokes should always be under tension. When a spoke is unloaded it can come loose or rub against the hub or other spokes. Even if it rubs or unscrews only a teeny tiny bit on each rotation it adds up. To ride 1000km you need about half a million rotations of the wheel.
Mixing spokes is a bad idea: The stronger spokes will stretch less, which contradicts the goal of distributing the load.
If the wheel is worth it, have it properly rebuilt (or learn to do so yourself, at this point you can’t make it worse). If the wheel is cheap it usually makes more sense to get a new one.
Without knowing your bike, that's a classic sign of a cheap wheel or built with cheap spokes. The added weight of the kid's seat won't help, but the underlying cause is the wheel and its build.
Your fixes are to replace all the remaining original spokes, or to replace the entire back wheel (hub/spokes/rim) and transfer over the tyre/tube.
If the spokes all broke at the J bend it can be an indication of low spoke tension all round. However when your first spokes were replaced, the bike shop should have tensioned them all better. Tightening all the spoke nipples equally may alleviate the problem or at least slow it down.
Personally I dislike the rear child seats. The ones where the kid sits over the top tube are much nicer for the rider and the child.