There is no trivial way to this because the PDF specification does not support it, except for some of the pre-defined Stamp annotations.
So you would have to make a custom appearance (and there are a few different ways to do this) but if later on someone were to resize, or make any changes (except to move) to the annotation, then you would lose the custom appearance and the default (if any) would appear.
Perhaps you could elaborate on what exactly you want to accomplish.
Could you provide a screenshot/image of an example?
Where does the image come from?
Where does the text come from?
Do you expect anyone reading the PDF to be able to resize/move/edit the annotation?