All perfect praise be to Allah, The Lord of the worlds. I testify that there is none worthy of worship except Allah and that Muhammad, sallallaahu ‘alayhi wa sallam, is His slave and Messenger.
The mosques are the houses of Allah Almighty, and they are the dearest places to Him. The Prophet, sallallaahu ‘alayhi wa sallam, said, “The parts of the land dearest to Allah are the mosques.” [Muslim] The Prophet, sallallaahu ‘alayhi wa sallam, also said, “Whoever performs ablution well in his home and thereafter comes to the mosque is the guest of Allah, and it is a duty upon the host to honor his guest.” [At-Tabaraani]
Therefore, the supplications are more likely answered in mosques than in other places. Shaykh Ibn ‘Uthaymeen said about supplication in mosques, “As for the places where supplications are likely to be answered, mosques are the places where the supplications are most likely to be accepted by Allah compared to other places...” He also said about supplicating in Al-Masjid Al-Haraam and Al-Masjid An-Nabawi, “In fact, the supplications recited in mosques are most likely to be answered, especially in the three sacred mosques: Al-Masjid Al-Haraam, Al-Masjid An-Nabawi, and Al-Masjid Al-Aqsa...”
As for asking the person setting off for Hajj or ‘Umrah to supplicate in one's favor, we do not know of an authentic hadeeth indicating that this was from the Sunnah of the Prophet, sallallaahu ‘alayhi wa sallam. There is a reported hadeeth to this effect; however, scholars of Hadeeth held different views regarding its authenticity. ‘Umar said, “I sought permission from the Prophet, sallallaahu ‘alayhi wa sallam, to perform ‘Umrah, and he granted me leave and said, ‘Oh brother, include us in your supplications and do not forget us.’” [Abu Daawood, At-Tirmithi, and Ibn Maajah - At-Tirmithi graded it hasan saheeh (sound or good)]
Based on this hadeeth, some scholars held that it is recommended to ask the pilgrim to supplicate Allah in one's favor. An-Nawawi said:
“It is recommended to bid farewell to the person setting off for pilgrimage, supplicate in his favor, and ask him to include you in his supplications, as we mentioned earlier and as evidenced by the hadeeth of ‘Umar in which he said, ‘I sought permission from the Prophet, sallallaahu ‘alayhi wa sallam, concerning ‘Umrah. He granted me permission and said, 'Oh my brother, do not forget to include us in your supplication.'’ ‘Umar commented, ‘He said to me a word that I would not be pleased to exchange for the whole world!’ Another version reads, ‘He, sallallaahu ‘alayhi wa sallam, said, 'Include us, my brother, in your supplications.' [Abu Dawood and At-Tirmithi - At-Tirmithi graded it hasan saheeh].'” [Al-Majmoo’]
Some scholars, though, classified this hadeeth as inauthentic given that its chain of narration includes ‘Aasim ibn ‘Ubaydullah, who was declared a weak reporter by Ibn Ma‘een, and Al-Bukhari and other scholars of Hadeeth declared him Munkar (his narration is rejected).
Allah knows best.