Her paternal aunt refuses to speak to her; what should she do?
Question

                Basically my aunt (dad's sister) who we were ever so close to and had a good relationship with has suddenly decided that my mum has said something about her mum. Now, she has stopped talking to my mum and ignores me and my sister. The thing is my mum hasn’t said anything and we have tried to clarify that but she is not having none of it. My question is if I see her, do I still call her? Do I still try to make that effort? Even if she doesn’t respond?! Or should I leave it and not talk to her as I know how she really feels about us and that she doesn’t speak to my mum. 
 I do feel it’s right that I do try on my side but then I feel like that’s not fair on my mum. Why do people always treat her bad and she doesn’t even say anything to anyone. Even if they always treat her bad. I’m really not sure where this leaves me. Is it right of me to keep trying to talk and even if I get ignored 1000x I should always try. My mum wouldn’t mind me giving salaam etc to them if I see them as she not a bad person at all. But would that be the right thing to do as they treat my mum so badly?
Praise be to Allah.Firstly:
Your upholding the ties with your paternal aunt, even though she is cutting you off, is truly upholding the ties of kinship as urged by Islamic teachings.
It was narrated from ‘Abdullah ibn ‘Amr that the Prophet (blessings and peace of Allah be upon him) said: “The one who upholds ties is not the one who reciprocates in kind; rather the one who upholds ties is the one who, if his relatives cut him off, he upholds ties with them.” Narrated by al-Bukhaari (5991).
Ibn al-Jawzi (may Allah have mercy on him) said:
What is meant by reciprocating in kind means reciprocating a deed with one that is similar.
The one who upholds ties of kinship for the sake of Allah, may He be exalted, does so as a means of drawing closer to Him, and in obedience to His command, even if they cut him off.
But if he upholds ties with them when they upholds ties with him, that is like paying off a debt.
Similarly, he said: “The best of charity is that given to a hostile relative.” That is because the motive for spending on a relative who is friendly and loving may be mixed with one’s whims and desires, but when spending on one who is hostile there is definitely no element of whims and desires mixed with the motive.
End quote from Kashf al-Mushkil (4/120-121).
Upholding the ties of kinship with a paternal aunt comes under the heading of obligatory upholding of the ties of kinship.
Shaykh al-Islam Ibn Taymiyah (may Allah have mercy on him) said:
Upholding the ties of kinship is obligatory according to scholarly consensus.
End quote from Majmoo‘ al-Fataawa (19/186).
The Muslim should remember that if he upholds ties with his relatives  even though they cut him off, he will be rewarded and they are sinning; he is in an honourable position of which the relatives are aware, even if they conceal that, and Allah, may He be exalted, will help him against them.
It was narrated from Abu Hurayrah that a man said: O Messenger of Allah, I have relatives with whom I try to keep in touch, but they cut me off. I treat them well, but they abuse me. I am patient and kind towards them, but they insult me.
He said: “If you are as you say, then it is as if you are putting hot ashes in their mouths. Allah will continue to support you as long as you continue to do that.
Narrated by Muslim (2558).
An-Nawawi said:
… What is meant is that it is as if you are feeding them hot ashes! This is likening what they feel of pain and anguish to what one who eats hot ashes feels of pain, and there is no blame on this doer of good; rather they incur great sin for cutting him off and causing him trouble…
End quote from Sharh Saheeh Muslim (16/115).
Secondly:
If this paternal aunt refuses to speak to you, then this is not an excuse to cut her off; rather you should hasten to uphold ties with her and not stop doing that, unless upholding ties with her will cause you harm and trouble, such as if she responds to you every time with insults and curses, and hurtful speech. In that case, it is sufficient simply to greet her with salaam, lest you fall into the kind of shunning and forsaking that is forbidden.
It was narrated from Abu Ayyoob al-Ansaari that the Messenger of Allah (blessings and peace of Allah be upon him) said: “It is not permissible for a Muslim to forsake his brother for more than three days, each of them turning away from the other when they meet. The better of them is the first to greet the other with salaam.”
Narrated by al-Bukhaari (6077) and Muslim (2560).
Ibn Rushd (may Allah have mercy on him) said:
With regard to a man forsaking another man, then he decides to greet him, without saying anything else to him, and he avoids speaking to him; do you think that he is no longer to be regarded as harbouring resentment against his brother?
He – Ibn al-Qaasim – said: I heard Maalik say: If the other man always offends or upsets him (whenever they speak), then he not to be regarded as harbouring resentment against his brother.
Ibn al-Qaasim said: I think that if the other man does not usually offend or upset him, then he is not be regarded as free of resentment against his brother.
Imam al-Qaadi said: What is meant by the words of Maalik and Ibn al-Qaasim is that the Muslim is not to be regarded as harbouring resentment against his brother if his brother usually offends or upsets the one who initiated the greeting, and it does not do any harm to the latter if he refrains from speaking to the offender. And if the one who is being greeted does not usually offend or upset the one who is initiating the greeting, then the latter cannot be regarded as being free of resentment towards his brother until he speaks to him, for he has no excuse for refraining from speaking to him.
End quote from al-Bayaan wa’t-Tahseel (10/60).
You do not have to keep on speaking to her and greeting her with salaam every time you see her, if she ignores you. Rather it is sufficient for you, in that case, to try to speak to her until you realise that she is insisting on cutting you off. At that point, you no longer have to keep on trying with her, even though that is better and is preferable, in the hope that her heart will soften one day.
But if your aunt still speaks to you and does not cut you off – rather she only cuts off your mother – then you should speak to her and try to bring about reconciliation between her and your mother. You should remember that the Prophet (blessings and peace of Allah be upon him) gave a concession allowing a kind of lying for the sake of bringing about reconciliation between people, because of the benefits that result for that, and there is no mischief involved in it. So there is nothing wrong with you telling her: My mother said something nice about you, and the like, and you can say the same thing to your mother, in the hope that Allah may make you a means of bringing about reconciliation between them.
And Allah knows best.