If the Quran is recited once for both the son and the mother, does each of them receive the full reward, or half of it? Or is it better to recite a complete ختمة (recitation) for each of them, especially if the mother cannot recite well?

It is permissible to dedicate the reward of reciting the Qur'an or other acts of worship to a Muslim, whether living or deceased, and the one to whom it is dedicated will benefit from that reward. The determination of the dedicated reward is based on the intention of the person dedicating it.