On my last job a client we work with gave me 150 Dirhams. She said that she wanted to bring me a gift but did not know what to bring. I tried to refuse, but she insisted to the point that I felt I would hurt her if I did not accept, so I took the money promising that I would give it back to her in one way or another. I kept forgetting about it for a year until one day when I heard that her daughter wants to learn English, I seized the occasion and offered her a new book of English lessons that cost 350 Dirhams (more than twice the amount she gave me). Is this enough or I should give her cash money as she gave me?

Praise be to Allah, the Lord of the Worlds; and may His blessings and peace be upon our Prophet Muhammad and upon all his Family and Companions.

You did very well by returning the good to whoever does it to you.  Ibn Umar reported that the Prophet (Sallallahu Alaihi wa Sallam) said: "Whoever does some good to you compensate him/her. If you do not find something to compensate him/her with, then keep on asking Allah for them until you feel satisfied that you have compensated them."

However, this compensation is not compulsory upon you unless the giver set, that as a condition.  In that case, it (this gift) is considered as a conditional gift.  This means, a gift by which the giver intends to get a compensation from the taker.  The taker has the choice between keeping the gift to return it later on to the giver or give the latter something of the same value in compensation.

This opinion is stated in Fiqh books.

Therefore, if this woman did not give you the 150 Dirhams to get a compensation for them, as understood from your question, then you did not have to give her anything in return.  But doing so, is likable.  Anyway, what you gave her compensates fully and completely what you took from her.

On the other hand, know that your accepting the gift in the first place is unlawful if it was against a service the customer deserved or if the institution where you work forbids taking such gifts.

Allah knows best.