From ignorance comes the propensity to ramble on ...
bAbli is indeed a pet name in Bengali, I have not heard it as a formal name. In Bengal, the usually short names by which people are actually called have rarely any relation to the formal names they hold, and often the same person is called by a different name by everyone close enough to them (aunts/uncles/grandfathers/grandmothers in a joint family, or close friends of parents, for example). These names are called pet names, and are sometimes the same words which can be used as formal names, but more often are short onomatopaeic words (I am never sure when this is used as an explanation), or words which have the phonemes of more common words all scrambled up into a sweet sounding pattern. They have a meaning so far as one can ask why they sound sweet to the bengali ear, and what phonemese they historically arose from.
Some of the more common themes include names starting with bAb- (bAban, bAbA, bAblu etc. for men, and bAbli, bAbi etc. for women) or m-m- (mAmmA, mom etc. for women) which presumably are related to the words for father (bAbA) or mother (mA). Sometimes, e.g. bAbu, maNTu, mantu, mANTA, mou, there are other origins possible (and some of these like mou are words and are very possibly misclassified here by me), but the use may be influenced by the b- or m- sounds. Another common theme is words ending in flowing -l-, jingling -nk- -ng-, soft -nt-, or ringing -NT-, bringing in the onomatopaeic elements into play.
For meanings, bAblA is the name of a thorny tree, and may also mean mentally deranged (in a Casper-the-ghost sense). These may have influenced the name bAbli, but I do not think explain the common use of this pet name.
Sometimes, popular pet names of yesteryears are used as formal names today.
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