While writing a letter to the people of Makkah, Imaam Hasan Al-Basri (RA) advised them of the following 15 places in Makkah where duas are accepted:
1. Inside the Baitullah
2. At Multazam
3. In Arafat
4. In Muzdalifah
5. In Mina
6. While performing Tawaf
7. While performing Sa’ee
8. At the hill of Safa
9. At the hill of Marwa
10. At the well of Zam Zam
11. At Maqam of Ibraheem
12. Under the Meezaab or aqueduct of the Ka’bah.
13. Near the big Shaytan
14. Near the middle Shaytan
15. Near the small Shaytan
In some other narrations places and instances such as Rukne- Yamani, Mataaf (where tawaf is commenced), near Hajr Aswad and on first sighting the Baytullah are also mentioned.
Source: A Guide to Hajj by Maulana Yakub Ismail Patel Kawiwala
Source: http://haqislam.org/blessed-places-where-duas-are-accepted/