Asalaam Alaikum
it is a proven and established in islam to make dua in certian times and certian places if the person is able. there are reports about making dua between the adhan and iqamah, during the last third of the night, during a certian time on friday, during the night of qadr etc. among the reports about making dua in certian places it is mentioned, the mount arafah, near the kaaba etc.
instructions to make dua in such times and places are given to us because they have a different degree of acceptance than the others. in the light of quran and sunnah i will try to clarify, insha Allah.
EVIDENCES FROM SUNNAH (this is the most authentic relevant content i found)
Imaam Ibn Maajah -rahimahullaah- reported in his “Sunan” [The Book of the Rites of Hajj: chapter (5): The Virtue of the Supplication of the person performing Hajj: (no.2893)]:
- Muhammad ibn Tareef narrated to us, saying: Imraan Ibn Uyainah narrated to us: from Ataa ibn as-Saa·ib: from Mujaahid: from Ibn Umar: from the Prophet sallallaahu alayhi wa sallam, that he said:
The fighter in Allaah’s cause, and the person performing Hajj, and the person performing `Umrah are Allaah’s deputation: He called them, and they responded to Him; and they ask of Him, and He gives to them.
Shaikh al-Albaanee -rahimahullaah- said: “Hasan”; and he said in “as-Saheehah” (no.1820), having quoted a supporting narration reported by al-Bazzaar (no.1153) as a hadeeth of Jaabir: The hadeeth, when the two chains of narration are gathered is “Hasan”.
- Imaam Ibn Maajah -rahimahullaah- reported (no.2895): Aboo Bakr ibn Abee Shaybah narrated to us, saying: Yazeed ibn Haaroon narrated to us: from Abdul-Malik ibn Abee Sulaymaan: from Abuz-Zubayr: from Safwaan ibn Abdillaah ibn Safwaan, who said:
"I was married to a daughter of Abud-Dardaa·, so he came to her and he found Ummud-Dardaa·, but he did not find Abud-Dardaa·. So she said to him: "Do you intend to perform Hajj this year?" He said: "Yes" She said: "Then supplicate to Allaah for good for us, since the Prophet sallallaahu alayhi wa sallam used to say":
The supplication which a person makes for his brother in his absence is responded to. There will be an Angel by his head saying: 'Aameen' [O Allaah! Grant it], to his supplication, ‘and may you receive the like of it’
He said: Then I went out to the market place, and I met Abud-Dardaa·, and he narrated the like of that to me from the Prophet sallallaahu alayhi wa sallam.
Shaikh al-Albaanee -rahimahullaah- said: "Saheeh."
NOTE
there are some reports about the excellence of supplications at Al-Mulltazim. but i have considered not to include them here in detail because the scholars have not agreed upon it, as it is not reported as an act directly from the prophet(PBUH), but it is reported from the sahabah.
refer : Iltizaam between the corner and the door
from THE RIGHTS OF HAJJ AND UMRAH
by Shaik Nasiruddin Albanee
CONCLUSION
with so many corroborating evidences in the sunnah, we can undertsand that making supplication/dua during hajj and umrah has a higher chance of acceptance. all this factors are amplified by the fact that, masjid Al-Haram is the mosque in which prayers are unequalled by any other and the land/area around the mosque is also blessed.
let us remind ourselves of this following hadith ;
Narrated Abu Huraira:
The Prophet (ﷺ) said, "Allah says: 'I am just as My slave thinks I am, (i.e. I am able to do for him what he thinks I can do for him)
may Allah the might and sublime grant us mercy and guide us to the straight path.
ALLAH KNOWS BEST