Does a mother's intention to donate her male newborn to Al-Azhar, and then her forgetting about it and him pursuing another education, constitute a vow, and what is its expiation?

The intention of a vow is not considered a vow, because a vow requires a specific formulation uttered by the one making the vow. Allah has forgiven the Ummah of Muhammad (peace be upon him) for what their souls whisper or think about, as long as they do not act upon it or speak of it. Therefore, mental existence has no effect; rather, what matters is verbal or practical existence. Accordingly, what was mentioned in the question is not a vow, and the woman is not obligated to do anything regarding it.