Why does JSON.stringify not serialize non-enumerable properties?

It's specified in the ES5 spec.

If Type(value) is Object, and IsCallable(value) is false

If the [[Class]] internal property of value is "Array" then

    Return the result of calling the abstract operation JA with argument value.

Else, return the result of calling the abstract operation JO with argument value.

So, let's look at JO. Here's the relevant section:

Let K be an internal List of Strings consisting of the names of all the own properties of value whose [[Enumerable]] attribute is true. The ordering of the Strings should be the same as that used by the Object.keys standard built-in function.