The emerald-eyed star's husband invited fans to help, and was flooded with thousands of responses.
"Suggestions for names are very welcome guys. Something with "A"," the actor posted on Twitter, four days after the baby was born.
Now, Abishek, says: "Evenings are spent reading out all your suggestions in front the family and trying to decide."
The little girl has been nicknamed 'Beti B' in the meantime – a moniker offered by one of the actor's co-stars.
"Till we find a suitable name for the little lady a dear friend @ShahanaGoswami gave me a great one. We have Sr B and Jr B already so Beti B," he wrote, referring to himself and his father Amitabh Bachchan (pictured holding the baby above), another famous Bollywood actor.
Sr B, every inch the proud grandfather, is delighted that fans have taken such an interest in naming his granddaughter.
"It is so lovable that they do this and I wish to thank the efforts that they make," he wrote on his blog. "Soon a name shall come about and we shall share it with all."
The baby will reportedly be named at a private family ceremony in December. Food – exclusively vegetarian dishes – has apparently been ordered, and the date has been set, though it won't be revealed to the public.
The traditional Hindu baby naming ceremony, known as the Namakaran, is a important day for new parents and a cause for celebration for the whole family.
It is usually performed on the 12th day after the birth of the child, though it can be held as late as the little one's first birthday.
It takes place at home or in a temple, where the father of the child whispers the name in the new arrival's right ear.
The priest offers prayers to bless and protect, and the baby's horoscope is also blessed.
After the naming ritual is over, friends and relatives are invited to offer blessings, touching honey or sugar to the baby's lips.