Some sources have mentioned that the Imam's surname (Aba Abdellah) is because of the name of his infant child named Abdullah that was only six months old and was martyred in Karbala.[1]  There is of course a different viewpoint that such nicknames were used even if one did not have a child by that name,

for example the nickname of Imam Mahdi is Aba Saleh (the father of Saleh) or the nickname of Imam Ali is Abu Turab (the father of Turab) even though he did not have a child by the name of Turab.

[1] Ibn Nama Hilli, Muthirul-Ahzan, pg. 7; Majlisi, Biharul-Anwar, vol. 45, pg. 47.