The name Eyüp comes from Abu Ayyub al-Ansari, the companion and standard bearer of the Prophet Muhammad. He came to Constantinople with the Arab army during the first attempted Muslim conquest of the city, died and as his last request was buried there. Seven centuries later, during the conquest of Constantinople, the tomb was said to have been re-discovered by Ak Şemsettin the Sheikh of Mehmed II (quote from Wikipedia). The mosque is outside the city walls and as a result often overlooked by tourists, who do not know what they are missing. It is a very popular mosque and seems to be visited in particular by families with a boy who has just - I was told mostly the day before - had his circumcision (sünnet) in their midst, they parade in a fine uniform and are treated to sweets and lots of attention. Also high dignitaries liked to be buried near this very holy spot, hence there is a world of türbeler (grave monuments) or just grave stones around.
This mosque is one of the three holiest Islamic places in Istanbul (The other two are the Sultan Ahmed (‘Blue’) mosque, whose courtyard was, in Ottoman times, the starting place for the yearly pelgrimage caravan to Mecca; and the Chambers of the Holy Relics of Islam in the Topkapı Palace, where the Mantle of the prophet and many other for muslims venerable objects are kept.).
During Ottoman rule all new sultans had to go to the Eyüp mosque (in solemn procession, of course), to gird the sword of the Prophet and be installed as Caliph (= deputy or representative of Allah). A tradition that started with Suleiman I the Magnificent in 1520.