It's worth considering the JSON you want.
Assuming you have this class
class Person
{
    public string Name { get; set; }
    public PhoneNumber HomePhone { get; set; }
}
You want this serialized to JSON, like this
{ "Name":"Joe", "HomePhone": "555-555-555" }
But you are getting JSON, something like this
{ "Name":"Joe","HomePhone": {"Number": "555-555-555"} }
--
To see why this is so, consider that the property Number of Person is an object. JSON is going to expect at least a {} to wrap the object - or more to the point, a set of name/values within the {}.
If you really, really want the former JSON syntax you need to register a custom converter for the Person object so that you can convince it to serialize and deserialze as a string. (see code example below).
However, I'd recommend that you just accept that because PhoneNumber is an object it correspond to a name/value dictionary when serialized into JSON and accept that the JSON is going to look a little less 'clean' than you might ideally want.
FWIW here's a code example that achieves your original aim (not the recommended approach)..
class Person
{
    public string Name { get; set; }
    public PhoneNumber HomeNumber { get; set; }
}
struct PhoneNumber
{
    private string _number;
    public PhoneNumber(string number)
    {
        _number = number;
    }
    public override string ToString()
    {
        return _number;
    }
}
class PersonConverter : JavaScriptConverter
{
    public override IEnumerable<Type> SupportedTypes
    {
        get { yield return typeof(Person); }
    }
    public override IDictionary<string, object> Serialize(object obj, JavaScriptSerializer serializer)
    {
        Person person = obj as Person;
        if (person != null)
        {
            Dictionary<string, object> dict = new Dictionary<string, object>();
            dict["Name"] = person.Name;
            dict["HomeNumber"] = person.HomeNumber.ToString();
            return dict;
        }
        return new Dictionary<string, object>();
    }
    public override object Deserialize(IDictionary<string, object> dict, Type type, JavaScriptSerializer serializer)
    {
        if (dict == null)
            throw new ArgumentNullException("dict");
        if (type == typeof(Person))
        {
            // Deserialize the Person's single property.
            string name = (string)dict["Name"];
            string homeNumber = (string)dict["HomeNumber"];
            // Create the instance to deserialize into.
            Person person = new Person()
            {
                Name = name,
                HomeNumber = new PhoneNumber(homeNumber)
            };
            return person;
        }
        return null;
    }
}
class Program
{
    static void Main(string[] args)
    {
        PhoneNumber number = new PhoneNumber("555 555");
        Person joe = new Person() { Name = "Joe", HomeNumber = number };
        JavaScriptSerializer serializer = new JavaScriptSerializer();
        serializer.RegisterConverters(new JavaScriptConverter[] { new PersonConverter() });
        Console.Out.WriteLine(serializer.Serialize(joe));
    }
}