An affectionate diminutive of Babur used fairly often is Babu... As in "My sweet Babu" Peppermint Patty uses this endearment for Charlie Brown in the Peanuts comic strip. Baby tiger is the inference. Also the moniker of the Genie character in a 70s Hanna Barbera cartoon.
This name is also spelled as Bobur. Originally it does not belong to India but to the land of Central Asia. Bobur was the grand grand son of Tamerlane. After he couldn't regain his throne and claimed royal status, he left the Timur's Empire, soon conquering the northern part of India founding there Babur's Empire. Because he was grand grand son of Tamerlane (Timur) who was married to the Mogul clan and gained that Mogul status, historically he is known as Mogul descendant.