gistfile1.cs

[DataVersion(2)]
public MyClass : MonoBehaviour
{
  public float m_myFloatThatUsedToBeAnInt;
}

#if UNITY_EDITOR
static class DataUpdater
{
  [DataUpdaterFor(typeof(MyClass))]
  static void DeserializeOldData(SerializedObject oldData, SerializedObject newData)
  {
    //oldData is the old serialized data
    //newData will be populated by Unity with a "best guess" (similar to what unity does today)
    //it's up to you to set newData serialized data to be how you want it

    //this method would be called when the typetree between stored data and new class is different.
    //(i.e. you added or changed a field),  or when the DataVersion is different between the stored data
    //and the current code

    //Questions: do you think it's good enough to have this API only work on MonoBehaviour/ScriptableObject,
    //and not on arbitrary nested classes. (you could update those too, but the code for that has to be in the consuming
    //monobehaviour, and not in the custom class itself)
  }
}
#endif

How common of a problem is this? The few times I've needed to migrate data between types in Unity, I did it by directly monkeying with the YAML serialized data (of a .prefab or .scene). You could provide a YAML munging library and let people go nuts with it.