All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad, sallallaahu ‘alayhi wa sallam, is His slave and Messenger.
If a person makes a vow, he must fulfill it according to the nature of the vow he had made and it is not valid to exchange it or refrain from fulfilling it as Allaah ordered us to fulfill our vows. Allaah says (what means): {Then let them end their untidiness and fulfill their vows and perform Tawaaf around the ancient House."} [Quran 22:29]
Allaah praised His pious slaves and mentioned some of their good qualities; Allaah says (what means): {They [are those who] fulfill [their] vows and fear a Day whose evil will be widespread.}[Quran 76:7]
Your sister offering the value of one of the goats she vowed to slaughter instead of slaughtering a goat is not sufficient as long as there are goats to slaughter even at a higher value. Therefore, your sister must fulfill her vow which is to give a goat in charity until her son is cured because the cure is the purpose that she defined in the formula of the vow.
However, the intention is given consideration in the vow as it is the case in an oath; so if, for example, she had intended to give charity in hope that the charity would be a cause for her son’s cure - and so the cure was the motivation for the charity - or according to what the doctors initially said before the loss of such hope, and then the doctors finally decided that his condition was hopeless, then in this case, she is not obliged to do anything after the doctors issued a report about his condition and that there is no hope for recovery.
In any case, her offering the value of the goat at the time of fulfilling the vow, i.e. before the report was issued, is not sufficient because she vowed to slaughter. If she was unable to slaughter, then she has to expiate for breaking her vow for each month she failed to fulfill the vow. For more benefit on expiation, please refer to Fatwa 82135.
Finally, it should be noted that your sister should not have made this vow as she could have given charity without making a vow, hoping that the charity would be a cause for the recovery of her child, as in the Hadeeth narrated by at-Tabaraani and al-Bayhaqi – which is a Hasan (good) Hadeeth – that the Prophet sallallaahu ‘alayhi wa sallam, said: “Treat your patients with charity."
Allaah Knows best.