Does anything become binding on my mother, in a Shar'i sense, due to the presence of the remaining Jamarat pebbles in her abaya pocket after her return from Hajj?

Scholars have unanimously agreed that throwing the pebbles (jamarat (definition)) with seven stones is from the guidance of the Prophet, peace and blessings be upon him. However, scholars have differed regarding the ruling on one who throws fewer than the required number of pebbles. The Shafi'is and Hanbalis are of the opinion that a sacrificial animal (dam) becomes obligatory for one who omits throwing all the pebbles, or throws for only one or two days, or omits three or more pebbles. According to the Shafi'is, for one pebble omitted, a "mudd" (a measure of food) is due, and for two pebbles, two "mudds" are due. As for the Hanafis, a "dam" is obligatory if the pilgrim omits throwing all the pebbles, or omits throwing for a full day, or omits throwing most of the pebbles for a day. If fewer are omitted, then a charity (sadaqa) is due. The Malikis made the "dam" obligatory for omitting even one pebble or for omitting all of them.

In summary, if you are certain that she omitted three or more pebbles from a single jamra, she must appoint someone to slaughter a sheep on her behalf in Mecca and distribute it among the poor of the Haram (definition). If she is not certain, then nothing is required of her.