As per Mahabharata, it was Brahma who suggested Vyasa to take help of Ganehsa. Veda Vyasa had conceived the whole poem in His mind but he was in anxiety regarding how to propagate it to his disciples. Brahma seeing Vyasa's anxiety came to Him and said:
I esteem thee for thy knowledge of divine mysteries, before the whole body of celebrated Munis distinguished for the sanctity of their lives. I know thou hast revealed the divine word, even from its first utterance, in the language of truth. Thou hast called thy present work a poem, wherefore it shall be a poem. There shall be no poets whose works may equal the descriptions of this poem, even, as the three other modes called Asrama are ever unequal in merit to the domestic Asrama. Let Ganesha be thought of, O Muni, for the purpose of writing the poem.
The Mahabharat doesn't give any specific reason why Ganesha was thought of for writing and not any other god, but Veda Vyasa did thought of Ganesha as he is the remover of obstacles and then narrated him the poem to be written:
Then Vyasa began to call to mind Ganesha. And Ganesha, remover of obstacles, ready to fulfill the desires of his votaries, was no sooner thought of, than he repaired to the place where Vyasa was seated. And when he had been saluted, and was seated, Vyasa addressed him thus, 'O guide of the Ganas! be thou the writer of the Bharata which I have formed in my imagination, and which I am about to repeat.
Taking the help of lord Ganesha for writing was certainly faster than the time Vyasa would have taken to write the whole poem Himself. It is because, lord Ganesha's pen didn't even stop writing for a moment apart from some difficult verses that Vyasa intentionally narrated. So apart from any other reasons, faster completion of such a huge work is certainly one reason why Veda Vyasa took the help of Ganesha, the god of knowledge and education.
Mahabharat, Adi Parva, 1