What is the ruling on shunning one's mother and refusing to sit with her at the dining table, because she went out to buy a toy for her grandson, even though her son had sworn to her not to go out, given his illness and her exhaustion and old age?

You must repent to Allah Almighty for saying you would not speak to or sit with your mother, because that is an offense to her. You are also required to offer an expiation (kaffarah) for breaking an oath, which becomes obligatory upon the one who swore the oath if they break it. Ibn Qudamah said: "If one says, 'By Allah, so-and-so will certainly do such-and-such, or will not do it,' or if one swore an oath upon a present person, saying, 'By Allah, you shall certainly do such-and-such,' and then caused that person to break the oath, and they did not do it, then the expiation is upon the one who swore the oath." The oath is broken if she does what you swore she would not do.