Praise be to Allah.
Firstly:
When exchanging money or currencies, when the currency is of the same type, it is stipulated that the amounts should be equal and the exchange should take place on the spot.
If the currencies are different, then it is stipulated that the exchange should take place on the spot only (the amounts do not have to be equal).
The basic principle concerning that is the words of the Prophet (blessings and peace of Allah be upon him): “Gold for gold, silver for silver, wheat for wheat, barley for barley, dates for dates, salt for salt, like for like, same for same, hand to hand. But if these commodities differ, then sell as you like, as long as it is hand to hand.” Narrated by Muslim (1578).
Currencies come under the same rulings as gold and silver.
It says in a statement of the Islamic Fiqh Council belonging to the Organisation of Islamic Cooperation (OIC):
With regard to rulings on paper currencies: they are promissory notes, but they come under the same rulings as gold and silver with regard to riba, zakaah, salam (payment in advance) and all other rulings.
End quote from Majallat al-Majma‘, issue no. 3, vol. 3, p. 1650 and issue no. 5, vol. 3, 1609.
It says in Fataawa al-Lajnah ad-Da’imah (13/459): A man has 500 riyals, and he wants to exchange it [for smaller banknotes], but the grocery storekeeper only had 300 riyals, and he will take the rest later on. Someone else objected to that and said that it is a kind of riba. We hope that you can advise us, may Allah bless you. As-salaamu ‘alaykum wa rahmat-Allahi wa barakaatuhu.
Answer: It is not permissible for two who are exchanging currency to part until each of them has taken the amount in full.
Based on that, it is not permissible for the one who gave someone five hundred riyals to exchange it (for smaller banknotes) to take three hundred now, and take the rest sometime after they part, even if it is only a short time.
And Allah is the source of strength. May Allah send blessings and peace upon our Prophet Muhammad and his companions.
Permanent Committee for Academic Research and Iftaa’.
Bakr Abu Zayd, ‘Abd al-‘Azeez Aal ash-Shaykh, Saalih al-Fawzaan, ‘Abdullah ibn Ghadyaan, ‘Abd ar-Razzaaq ‘Afeefi, ‘Abd al-‘Azeez ibn ‘Abdillah ibn Baaz. End quote.
Secondly:
There is nothing wrong with appointing someone else to do the exchange, even if he does it with himself, so that he plays the role of both parties, taking from himself on behalf of the one who appointed him to do that.
Ibn Qudaamah said in al-Mughni (4/41): If he wants to exchange with him ten dirhams for one dinar, and he gives him more than one dinar so that he can take it back from him at a later time, that is permissible, even if he takes it after a long time; the extra amount becomes a trust that is in the other person’s keeping, and he is not liable if it gets damaged or destroyed (through no fault of the one who is keeping it for him). End quote.
It says in Mataalib Ooli an-Nuha (3/151): That is like the case when someone is appointed to take from himself and exchange with himself, which is permissible. Thus he plays the role of both parties in the transaction. End quote.
Based on that, if you gave your mother permission to do this exchange, and you both agreed to exchange that banknote for her from your own wealth, then there is nothing wrong with what she did of doing the exchange herself, and playing the role of both parties in the transaction.
And Allah knows best.
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