>I've got an AddProperty() in a button class INIT method. Where do I set it's value in the subclasses to make sure it is populated at runtime? Want to make sure the horse is in front of the cart. Thanks!

Your could use the same AddProperty to set its value:

this.AddProperty([NewOne], some value)

That way you would have default value for it.
But if you want to change it in some subclass you should do this programatically it Init event of the subclass:

*** Init:
DODEFAULT() && That is mandatory. If you don't use that your property will not be created.
this.NewOne = other value

Why not add it with Class menu and make that property persistent for all?