West Bengal has “West” in its name for a historical reason. Until India gained independence from the British in 1947, Bengal was a single, giant entity under British administration. After the partition of India, Bengal was itself divided along religious lines into two separate entities: East Bengal, which joined Pakistan (and eventually became Bangladesh), and West Bengal, which remained part of India.

At that time, to differentiate between the two regions, the western part of Bengal locally within India was named “West Bengal” and the eastern part became “East Bengal”. East Bengal is what is today Bangladesh and in that geographical context it made sense but West Bengal’s name stayed on.
West Bengal is called so to maintain the historical as well as cultural identity. Though West Bengal falls in the eastern part of India, but ‘West’ denotes it’s position with respect to the old undivided Bengal during partition. And it has become a kind of an identity of the state.