The month of Ramadaan is a blessed month, during which the Qur'aan was revealed, and the process of inspiration started upon the Prophet صلى الله عليه وسلم. Therefore, the du'a of Ramadaan is a blessed du'aa', and this can be inferred from the Prophet's hadeeth صلى الله عليه وسلم: 

إِذَا دَخَلَ رَمَضَانُ فُتِّحَتْ أَبْوَابُ الجَنَّةِ ، وَغُلِّقَتْ أَبْوَابُ جَهَنَّمَ ، وَسُلْسِلَتِ الشَّيَاطِينُ 

"When Ramadaan comes, the Doors of Paradise are opened, and the Doors of Hell are closed, and the Shayaateen are chained up." [1] 

So it can be inferred from this hadeeth that du'aa' during Ramadaan has a greater chance of being accepted, as the Gates of Paradise and Mercy are opened. [2] 

Notes: 

[1] Reported by al-Bukhaari and Muslim. 

[2] Shaykh Dr. Abu Ammar Yasir Qadhi حفظه الله, Du'a - The Weapon of the Believer, p. 128.

See also: 

Fasting the Month of Ramadaan

Dua (Supplication)