Masjid al-Ijabah
Names
It is called Masjid al-Ijabah because the Prophet (Allah bless him and give him peace) supplicated to Allah Most High in this location and requested three things, two of which he was granted and the third which he was not. Historians mention that because of this, it was named Masjid al-Ijabah (The Mosque of the Answering). It has also been called Masjid Bani Muawiyah due to its location in the district of the Bani Muawiyah of the Ansar (Al-Samhudi mentioned this in Wafa al-Wafa).
Location
The Prayer and Supplication
It is established that the Prophet (Allah bless him and give him peace) prayed in Masjid al-Ijabah and supplicated therein as narrated in Sahih Muslim,
"The Messenger of Allah (Allah bless him and give him peace) came one day from al-Aliyah and passed by Masjid Bani Muawiyah. He entered the Masjid and offered two units of prayer which we offered with him. He then supplicated to his Lord for a lengthy period of time, then turned to us and said: I asked my Lord for three matters, He granted me two and prevented me from one. I asked my Lord that my community not be destroyed by famine which I was granted. I asked Him that my community not be destroyed by flooding and I was granted it. I asked Him that He protect them from internal strife but I was prevented from this."
Imam Malik narrated that Abdullah bin Umar visited Masjid al-Ijabah after the passing of the Prophet (Allah bless him and give him peace) and asked regarding the spot where the Prophet (Allah bless him and give him peace) prayed and the three supplications which were made, commenting on the last supplication that,
"Discord will remain until the day of judgement."
Ibn Shabbah narrates in his Tarikh al-Madina from Ghassan bin Muhammad bin Talha that,
"It has reached me that the Prophet (Allah bless him and give him peace) prayed in Masjid Bani Muawiyah a distance of 2 cubits to the right of the mihrab."
Ibn Shabbah also narrates from Sa's bin Abi Waqqas that he was with the Prophet (Allah bless him and give him peace) when he passed by the Masjid Bani Muawiyah. He entered it, offered two units of prayer and then stood supplicating to his Lord, after which he left.
Imam al-Samhudi commenting on these narrations states,
"A person should seek out to pray in this location, and that they supplicate standing up after praying due to the preceding narrations."
Note: This is in reference to the nafl prayer.