I saw several times on Pinterest that you can take a permanent marker, write on a dollar store plate and bake it for 30 minutes and it would be permanent. Well, I tried it and it's just not true. It wipes off immediately. After some trial and error, I've found what does work though. Oil based paint pens. There are a couple of different brands. I found some at Hobby Lobby and at Joann's. Make sure they're non-toxic.
I traced my kids hands on cardstock then cut them out to use as stencils. Use the paint pens to mark on the dollar store bought dishes. Bake at 350° for 30 minutes directly on the rack. Turn off oven and let cool in oven. Remove and it should be permanent...mine are. Although I'd recommend gently hand washing still. When I ran it through the dishwasher, it came off in places.
For the mug, I traced the hands on both sides of the cup so that it kind of looks like they're holding the cup too.
You can also put the kids names on the bottom of the dish and the date or other note.