myObject.doSomething = MyClass.prototype.myFunction
JSON obviously does not hold onto the functions themselves is only stores simple typed variables. The way I have addressed this in the pass is to be a restore method in my class and simply call that method with the data from JSON so as to re-populate the class with the data that belongs in it.
I have done this extensively with the Value Object ( VO ) design pattern in my code base and it has worked quite well for me. Just a word of a caution though, Ie7/Ie8 are not terribly friendly with this approach if you try to communicate across windows. As I recall I think it is IE7 that does not return the right "typeof" for some properties so I ran into a whole bunch of challenges in my restore when cross-window communication was involved.