Interesting, (I may be wrong) Theoratically, an existing class cannot be extended with new members at runtime. However, new class can be created using System.Reflection.Emit that has the existing class as base class. But not sure it will also work at intialization or not. I am wondering, Simple Inheritance usually used to extend some model.
https://msdn.microsoft.com/en-us/library/system.reflection.emit.aspx

/K

In an ideal world you sould update your model source code but, in situations where you can't access that model, you could create something as an initialization module as you suggested. Something like this should do it.

[InitializableModule]
[ModuleDependency(typeof(EPiServer.Web.InitializationModule))]
public class ModelInitialisation : IInitializableModule
{
    public void Initialize(EPiServer.Framework.Initialization.InitializationEngine context)
    {
        var typeRepo = ServiceLocator.Current.GetInstance<IContentTypeRepository>();
        var contentType = typeRepo.Load("MyContentType");
        var pdr = ServiceLocator.Current.GetInstance<IPropertyDefinitionRepository>();
        if (!contentType.PropertyDefinitions.Any(d => d.Name.Equals("MyNewPropertyName")))
        {
            var dtr = ServiceLocator.Current.GetInstance<IPropertyDefinitionTypeRepository>();
            var tdr = ServiceLocator.Current.GetInstance<ITabDefinitionRepository>();
            var def = new PropertyDefinition();
            def.Name = "MyNewPropertyName";
            def.ContentTypeID = contentType.ID;
            def.EditCaption = "My New Property Name";
            def.Type = dtr.Load(typeof(PropertyString));
            def.Searchable = false;
            def.Required = false;
            def.LanguageSpecific = true;
            def.FieldOrder = 100;
            def.Tab = tdr.Load(SystemTabNames.Settings);
            pdr.Save(def);
        }
    }

}

@Paul, this works pretty well -- any idea how to specify a particular editordescriptor or clienteditingclass for this dynamically added property definition?