All perfect praise be to Allah, The Lord of the Worlds, and may His peace and blessings be upon our Prophet Muhammad and upon all his family and companions.
There is no clear text proving that a woman's supplication while giving birth is likely to be answered. Probably, pains that she endures make her sincere in supplicating Allah, and this is why people tend to think so. In this regard, Allah, The Almighty, Says (what means): "Is not He (best) Who answereth the wronged one when he crieth unto Him and removeth the evil." [An-Naml/27].
Moreover, whoever supplicates Allah, The Exalted, in sincerity and humility, his supplication is likely to be answered in one of the following three forms, where the Prophet (PBUH), said (what means): "Whenever a Muslim supplicates Allah, He accepts his supplication, compensates him in the hereafter or averts any similar kind of trouble from him until he prays for something sinful or something that may break the ties of kinship." Upon this someone of the Companions said: "Then we shall supplicate plenty." The Messenger of Allah (PBUH) said: "Allah is more plentiful (in responding)." [Ahmed, vol.3/pp.18] and [Al-Tirmithi, pp.3573].
Therefore, if the supplication involved something sinful or something that may break the ties of kinship then Allah won't answer it, and a Muslim isn't permitted to supplicate for such matters. And Allah Knows Best.