If there is only one relative, he takes the whole estate
Question

                I have an elderly female relative who has only one heir, the son of her only full sister who is deceased, and that was after checking all her relatives. So she wants to know what his share of the estate will be and whether it is permissible for her to bequeath the rest of the estate to the person who raised him, as she does not have any children.
Praise be to Allah.
    The son of one’s full sister is a dhu rahm relative (a 
    relative on the mother’s side) and the scholars differed concerning their 
    inheritance. The correct view is that they may inherit if there is no one 
    who is entitled to a share of inheritance (fareedah) or relatives on 
    the father’s side (‘usbah). 

    Among those who were of the view that they may inherit were: 
    ‘Ali, Ibn Mas’ood, Ibn ‘Abbaas according to the most well known report from 
    him, Mu’aadh ibn Jabal, Abu’l-Darda’, and Abu ‘Ubaydah ibn al-Jarraah (may 
    Allaah be pleased with them). This is the view of the Hanafis and of Imam 
    Ahmad, and the later Maalikis and Shaafa’is. 

    See: al-Mawsoo’ah al-Fiqhiyyah (3/54). 

    Shaykh Ibn ‘Uthaymeen (may Allaah have mercy on him) said: 
    The scholars (may Allaah have mercy on them) differed with regard to the 
    inheritance of dhu rahm relatives, but the most correct view is that 
    they must inherit, because the Prophet  (peace and blessings of Allaah 
    be upon him) said: “The maternal aunt is of the same status as the mother.” 
    And he also said: “The maternal uncle is the heir of the one who has no 
    heir.” And Allaah says (interpretation of the meaning): “But kindred by 
    blood are nearer to one another (regarding inheritance) in the decree 
    ordained by Allaah” [al-Anfaal 8:75]. The view that they do not inherit 
    is a weak view. How can we deprive the maternal uncle or the mother’s father 
    of their relative’s wealth then put it in the bayt al-maal to be consumed by 
    people who are not related at all? 

    Islam would not decree such a thing. The definitely correct 
    view is that dhu rahm relatives do inherit, but only if there 
    is no heir who is entitled to a fareedah share and no relative on the 
    father’s side. End quote from al-Sharh al-Mumti’ (5/73). 

    With regard to how the dhu rahm relatives inherit, the 
    one who is present takes the place of his immediate ascendant, and takes his 
    share of the estate. In the case mentioned in the question, the sister’s son 
    takes the place of the sister, so he takes her share, which is half. Then 
    the rest of the estate goes to him too, so he gets the whole estate, because 
    there is no other heir.  

    It says in Kashshaaf al-Qinaa’ (4/456): If there is 
    only one dhu rahm relative, he takes all the wealth, because he takes 
    the place of his immediate ascendant.  

    See: al-Mawsoo’ah al-Fiqhiyyah (3/54). 

    Based on this, if this woman dies, and she does not have any 
    heir apart from her sister’s son, then he will take the entire estate. 

    She has the right to bequeath no more than one-third to the 
    person who raised him, or to anyone else, because of the report narrated by 
    al-Bukhaari (5659) and Muslim (1628) that Sa’d ibn Abi Waqqaas (may Allaah 
    be pleased with him) said to the Prophet  (peace and blessings of Allaah 
    be upon him): O Prophet of Allaah, I am going to leave wealth behind, and I 
    am only leaving behind one daughter. Can I bequeath two-thirds of my wealth 
    and leave her one-third? He said: “No.” He said: “Can I bequeath half and 
    leave her half? He said: “No.” He said: Can I bequeath one-third and leave 
    her two-thirds? He said: “One third, and a third is a lot.” 

    The Messenger  (peace and blessings of Allaah be upon 
    him) allowed him to bequeath one third of his wealth, and the scholars are 
    agreed that it is not permissible to bequeath more than one third. 

    See: al-Mughni (8/404). 

    And Allaah knows best.