He is not sure about the meat that his mother buys

Praise be to Allah.

Firstly: 

If the meat has been slaughtered by one of the People of the Book (a Jew or a Christian) and we do not know the method of slaughter, but most people in the country kill meat by means of electric shock or by slaughtering it for someone or something other than Allah, it is not permissible to eat that meat, on the basis of what is usually the case. The same applies if one is not sure whether the meat was slaughtered properly or killed by means of electric shock. That is because of the report narrated by al-Bukhaari (5485) and Muslim (1929) from ‘Adiyy ibn Haatim (may Allah be pleased with him) from the Prophet (blessings and peace of Allah be upon him) who said: “If you release your (hunting) dog and say the name of Allah, and (the dog) catches (the game) and kills it, then eat, but if (the dog) ate (some of it), then do not eat, because he only caught it for himself. If another dog joins him over whom the name of Allah was not mentioned, and they caught it and ate from it, do not eat, for you do not know which of them killed it. If you shoot an arrow and you find (the game) after one or two days with nothing but the mark of your arrow on it, then eat, but if it fell into water, do not eat it.” 

According to a report narrated by Muslim: “When you shoot your arrow, mention the name of Allah. Then if you find it (the game) dead then eat, unless you find that it has fallen into water, in which case you cannot know whether it was the water that killed it, or your arrow.” 

The Prophet (blessings and peace of Allah be upon him) forbade eating if there is any uncertainty as to whether the meat was slaughtered properly in accordance with sharee‘ah because of the hunting dog being joined by another dog, or because of the target animal falling into water and the possibility that it died by drowning. 

Al-Nawawi (may Allah have mercy on him) said in Sharh Muslim: The words of the Prophet (blessings and peace of Allah be upon him): “If you find with your dog another dog and it joined in killing (the game), Do not eat because you do not know which of them killed it.” This states an important principle, which is that if there is any doubt as to whether the animal was slaughtered in the permissible manner, it is not halaal, because the basic principle is that it is haraam. This is a point on which there is no difference of scholarly opinion. End quote. 

Al-Haafiz Ibn Hajar said in Fath al-Baari: The words “if it fell into the water, do not eat it” – the reason for the prohibition on eating it is to be taken from the phrase above, because in that case there is doubt as to whether it was killed by the arrow or by drowning in water. If it is established that the arrow struck it and it died, and it did not fall into the water until after it was killed by the arrow, then it is permissible to eat it. al-Nawawi said in Sharh Muslim: If the game is found in the water, drowned, it is haraam according to consensus. End quote. al-Raafi‘i said that if he reaches it in time (before it dies) and cuts its throat for example, then he has slaughtered it in the proper manner. His opinion is supported by the report narrated by Muslim, “you cannot know whether it was the water that killed it, or your arrow.” This indicates that if he knows that his arrow is what killed it, then it is permissible. End quote. 

For more information please see the answer to question no. 10339. 

Secondly: 

If your mother or someone else brings haraam meat, it is not permissible to eat it, because Allah, may He be exalted says (interpretation of the meaning):

“Forbidden to you (for food) are: Al-Maitah (the dead animals - cattle - beast not slaughtered), blood, the flesh of swine, and that on which Allâh’s Name has not been mentioned while slaughtering, (that which has been slaughtered as a sacrifice for others than Allâh, or has been slaughtered for idols) and that which has been killed by strangling, or by a violent blow, or by a headlong fall, or by the goring of horns - and that which has been (partly) eaten by a wild animal - unless you are able to slaughter it (before its death) - and that which is sacrificed (slaughtered) on An-Nusub (stone altars). (Forbidden) also is to use arrows seeking luck or decision; (all) that is Fisqun (disobedience of Allâh and sin).”

[al-Maa’idah 5:3]. 

You have to be gentle and polite when explaining why you do not want to eat this meat, and try to make your mother happy by eating other food that she offers; bring her some halaal meat to cook when you are there and try to convince her of the importance and benefits of eating halaal meat and the harmful effects of eating meat that was not slaughtered in the proper shar‘i manner. This is due to the rights that she has of respect and kind treatment. 

See also the answer to question no. 172

And Allah knows best.