Should he let his Christian aunt stay with him so that he can call her to Islam?

Firstly: 

It is permissible for you to let your aunt stay with you so that you can call her to Islam and encourage her with wisdom and beautiful preaching, If she sees you and your good attitude and kind treatment, that may be a means of opening her heart to this great religion. Attitude and actions may often be more eloquent than words and preaching. So strive hard to do that, and if Allaah guides her at your hands, that will be better for you than this world and everything in it.  

You should equip yourself with patience and forbearance, and respond to bad with good, and to mistakes with forgiveness, as Allaah says (interpretation of the meaning): 

“The good deed and the evil deed cannot be equal. Repel (the evil) with one which is better (i.e. Allaah orders the faithful believers to be patient at the time of anger, and to excuse those who treat them badly) then verily he, between whom and you there was enmity, (will become) as though he was a close friend”

[Fussilat 41:34]

One of the indications that it is permissible to let a kaafir stay among the Muslims, so that he may see their attitude and come to know their religion, is that which is narrated concerning the story of Thumaamah ibn Uthaal, who was tied up in the mosque for three days after the Muslims took him prisoner. Then the Prophet (peace and blessings of Allaah be upon him) pardoned him and he became Muslim. The story is narrated in Saheeh al-Bukhaari (4372) and Saheeh Muslim (1764). 

The aunt also has kinship rights because she is a close relative, even if she is a kaafir. The Prophet (peace and blessings of Allaah be upon him) said: “The maternal aunt is like the mother.” Narrated by al-Bukhaari, 2700. 

Al-Tirmidhi (3975) narrated from ‘Umar that a man came to the Prophet (peace and blessings of Allaah be upon him) and said: “O Messenger of Allaah, I have committed a grievous sin; can I repent?’ He said: “Do you have a mother?” He said, “No.” He said: “Do you have a maternal aunt?” He said: “Yes.” He said: “Then honour her.” Classed as saheeh by al-Albaani in Saheeh al-Tirmidhi. 

And he (peace and blessings of Allaah be upon him) said to Asma’: “Uphold the times of kinship with your mother.” Narrated by al-Bukhaari, 2620; Muslim, 1003. And her mother was a kaafir. 

Secondly: 

There are two basic conditions for a person to stay in a non-Muslim country, as stated by the scholars. Shaykh Ibn ‘Uthaymeen (may Allaah have mercy on him) said: 

1 – He should be certain that he is able to remain strongly committed to his religion, as he has enough knowledge, faith and determination to remain steadfast in his religious commitment and to avoid deviation and misguidance. He should bear enmity towards the kuffaar in his heart and avoid taking them as friends and liking them, because doing so is contrary to faith. Allaah says (interpretation of the meaning): 

“You (O Muhammad) will not find any people who believe in Allaah and the Last Day, making friendship with those who oppose Allaah and His Messenger (Muhammad), even though they were their fathers or their sons or their brothers or their kindred (people”

[al-Mujaadilah 58:22]  

“O you who believe! Take not the Jews and the Christians as Awliyaa’ (friends, protectors, helpers), they are but Awliyaa’ of each other. And if any amongst you takes them (as Awliyaa’), then surely, he is one of them. Verily, Allaah guides not those people who are the Zaalimoon (polytheists and wrongdoers and unjust).

52. And you see those in whose hearts there is a disease (of hypocrisy), they hurry to their friendship, saying: “We fear lest some misfortune of a disaster may befall us.” Perhaps Allaah may bring a victory or a decision according to His Will. Then they will become regretful for what they have been keeping as a secret in themselves”

[al-Maa’idah 5:51, 52]

And it is proven in al-Saheehayn that the Prophet (peace and blessings of Allaah be upon him) said: “A man will be with those whom he loves.” Narrated by al-Bukhaari, 6168; Muslim, 2641. 

Loving the enemies of Allaah is one of the most serious dangers that a Muslim faces, because loving them implies that one agrees with them and follows them, or at the very least that he is not denouncing them. 

2 – The second condition is that he should be able to practise his religion openly, in the sense that he can establish the rituals and symbols of Islam with no impediment, so he is not prevented from establishing regular prayer, and praying Jumu’ah and offering prayers in congregation if there are others with him who can pray in congregation and establish Jumu’ah; and he should not be prevented from paying zakaah, fasting, performing Hajj and doing all the other rituals of Islam. If he is not able to do that, then it is not permissible for him to stay there and he must migrate in that case. 

End quote from Sharh al-Usool al-Thalaathah by Shaykh Ibn ‘Uthaymeen (may Allaah have mercy on him), in Majmoo’ al-Fataawa, 6/132 

Whoever cannot meet these two conditions must migrate to a Muslim land if he is able to do so. If he is unable to migrate, then he is excused, because Allaah says (interpretation of the meaning): 

“Verily, as for those whom the angels take (in death) while they are wronging themselves (as they stayed among the disbelievers even though emigration was obligatory for them), they (angels) say (to them): “In what (condition) were you?” They reply: “We were weak and oppressed on the earth.” They (angels) say: “Was not the earth of Allaah spacious enough for you to emigrate therein?” Such men will find their abode in Hell — what an evil destination!

98. Except the weak ones among men, women and children who cannot devise a plan, nor are they able to direct their way.

99. These are they whom Allaah is likely to forgive them, and Allaah is Ever Oft-Pardoning, Oft-Forgiving”[al-Nisa’ 4:97-99]

See al-Mughni by Ibn Qudaamah, 8/457 

See also question no, 22309. 

And Allaah knows best.