The space in between the Hajr-e-Aswad and the door of Kaaba is known as “Multazam”. Al-Multazam is a part of the Ka‘bah that pilgrims adhere to for the purpose of supplicating and seeking refuge, and is located between the Black Stone and the gate. 
The multazam (place of clinging) is the part of the Ka’bah that is between the Black Stone and the door of the Ka’bah. What is meant by iltizaam (clinging) is when the supplicant (person making du’aa’) places his chest, face, forearms and palms against it and calls upon Allaah saying whatever du’aa’ he wishes.
It was narrated from ‘Amr ibn Shu’ayb that his father said: I circumambulated the Ka’bah with ‘Abd-Allaah, and when we came to the back of the Ka’bah I said: Will you not seek refuge with Allaah? He said: We seek refuge with Allaah from the Fire. Then he proceeded to touch the Stone, and he stood between the Corner and the door, and placed his chest, face, forearms and hands like this, and spread them out. Then he said: This is what I saw the Messenger of Allaahﷺdo. Shaykh al-Albaani classed it as saheeh in al-Silsilah al-Saheehah, 2138
There is no specific du’aa’ that the Muslim should say in that place. He can cling to the Multazam when he enters the Ka’bah (if it is easy for him to enter) or he may do that before performing the Farewell tawaaf (tawaaf al-wadaa’), or he may do it at any time he wants. He should not cause difficulty for other people by offering a lengthy du’aa’. Similarly it is not permissible to crowd other people or annoy them in order to cling there. If he sees a space then he should say du’aa’. Otherwise it is sufficient for him to say du’aa’ whilst circumambulating. 
Yahya related to me from Malik that he had heard that Abdullah ibn Abbas(R) used to say that the area between the corner of the Black Stone and the door of the Kaba was called al-Multazam. Muwatta Malik Book 20, Hadith 957 
Multazam is also called Al-Mad‘a and Al-Muta‘wwath.