There is no basis for Hijr Ismaa’eel being so called
First of all we should point out that there is no basis for the people calling Hijr Ismaa’eel by this name, and Ismaa’eel knew nothing of this Hijr. Ibraaheem and Ismaa’eel (peace be upon them) built the Ka’bah completely, including the area of the Hijr, then the walls of the Ka’bah became weakened because of a fire and flood that happened before the mission of the Prophet (peace and blessings of Allaah be upon him) began. So Quraysh knocked down what was left of its walls and rebuilt it. They ran out of money from good sources and were thus unable to complete the building on the foundations of Ibraaheem and Ismaa’eel (peace be upon them), so they left the Hijr outside, and built a small wall around it to indicate that it was part of the Ka’bah. They had stipulated for themselves the condition that nothing should be used for building it except money from good sources, and that the earnings of prostitutes or profits from interest-based sales could not be used, or money that was acquired wrongfully by anyone.
In al-Saheehayn it is narrated that ‘Aa’ishah (may Allaah be pleased with her) said: “I asked the Prophet (peace and blessings of Allaah be upon him) about the wall – is it part of the House? He said, ‘Yes.’ I said, ‘Why did they not incorporate it into the House?’ He said, ‘Your people ran out of money.’”
Narrated by al-Bukhaari, 1584; Muslim, 1333.
“The wall” here refers to the Hijr.
The correct way is to call it al-Hijr, without attributing it to Ismaa’eel (peace be upon him).
It is not proven in any marfoo’ hadeeth that Ismaa’eel (peace be upon him) was buried in this Hijr, or that Haajar was buried in it. But there are mawqoof reports with weak isnaads which may be understood as meaning that the grave of Ismaa’eel is inside the Hijr.
For more information see Tahdheer al-Saajid min Ittikhaadh al-Quboor Masaajid (Warning to Worshippers against Taking Graves as Places of Worship) by Shaykh al-Albaani (may Allaah have mercy on him), p. 75, 76.
It is extremely unlikely that Ismaa’eel (peace be upon him) would have buried his mother inside the Ka’bah, or that his sons would have buried him there. Such a suggestion needs proof, and no such thing has been proven, praise be to Allaah.
And Allaah knows best.