I imagine this is a common issue. Parents decide that it is time for their son to start wearing a yarmulke. He, being in the range of 2-4, doesn't recognize the value of having something on his head that falls off when he turns upside down. What are some effective strategies for getting a young child to be willing, or perhaps even excited, to wear his yarmulke?
You might want to take him to the store to pick out his very own yarmulke. Now it is his choice and he is doing it because he wants to do it. And every day he wears it he is wearing his yarmulke that he picked out.
I have found that it is always best to give children choices when the choices are limited to approved behaviors. Children thus feel empowered and no longer feel that they are simply taking orders from a parent but are making the decision themselves.
I don't think this is as big of a problem as you would think. If his father always wears a yarmulke, that alone could excite him: "You get to wear a yarmulke just like daddy does!".
With regards to it falling off, you could always add clips.
Just like with regards to everything in child-rearing, if the kid grows up in an atmosphere that honors Torah and Halachah, he is more likely to honor and respect the Torah and Halachah on his own. He is then more likely to return the yarmulke to his head on his own if it falls off.