What is the best time for making Du'a in Ramadan? 

Du'a in Ramadan is an essential matter. The Prophet, peace be upon him said:

(There is a time for the one who observes Siyaam at which his Du'a is accepted.)

    
According to a Fiqhi (jurisprudent) rule, when things are not specified, then they are non-finite, and thus the Du'a made in Ramadan is not restricted to a specific time. Man can make Du'a after he prays al-Asr (afternoon prayer), al-Fajr (dawn prayer) in Masjid, or Qiyam al-lail (night prayer). However, Allah the Almighty says:  

(…And recite the Qur'an in the early dawn (i.e. the morning prayer). Verily, the recitation of the Qur'an in the early dawn is ever witnessed (attended by the angels in charge of mankind of the day and the night).)

[Al-Isra', 78]

 
At dawn, man receives no calls and no unexpected guests; this time is dedicated for Allah, Exalted and Sublime be He.