Avoid posing your subjects
You'll get more natural expressions if you can compose shots where the subject is actively doing something, rather than just looking at the camera - and you may well find that these make better scenes for your memories or to share with your friends. So, if you can concentrate on photos of your child concentrating on riding a bike or constructing a model, you might be able to avoid the problem much of the time.
"Smile!" - Click - "Relax!" - Click (the real photo, without the cheesy grin). This works surprisingly well even when your subject has learnt what you're doing. Sometimes I even say that I'm going to take the "real picture" followed by a "fun shot" afterwards. The nice thing here is that when people relax after holding a pose, they tend to retain a natural version of a smile.
The favourite of family photographers everywhere, though largely dependent on whether you have a gift for comedy. I get quite good results with adults by setting them up to pose and then breaking the tension with a gently sarcastic comment, perhaps like, "No, look like you're enjoying this!". I expect some friendly teasing to work well for children, too.
This part is outside of my experience, but I think it might be possible to change the perception of the camera and get the subject to treat it more like one of their peer group rather than a very critical judge. Perhaps we can try exercising the child's imagination to pretend it's not a big scary black lens. Or, for less imaginative types, attach or hold a favourite toy near the camera (a bit like the old "watch the birdie") to redirect their attention?
With modern equipment, we have some advantages that early photographers didn't: there's no need to hold still for seconds at a time (although we do tend to carry this attitude still). With instant review and almost unlimited exposures, we can chimp the results together with our subjects and discuss which we prefer ("Shall we try another like this one?").
It's hard getting people to relax, and there's a risk that the harder you try, the more tense you and they will both feel. We need a stock of techniques to break the stiffness and let the subject move and breathe naturally.