It is named Bilaskhani Todi because of the Meend used while singing the notes 'Ga' and 'Re', a common feature of the Raag 'Todi'.
The Hindustani classical raga Bilaskhani Todi is an example of the flaws of the Bhatkhande thaat system because it is classified under the Bhairavi thaat based on the notes it uses, but it is actually a type of Todi, and permitting any Bhairavi during a performance kills the raga.
[1] Thaat: Bhairavi Morning, between 6 a.m. to 12 p.m. Certain ragas have seasonal associations.
Devotional, Bhakti Legend has it that this raga was created by Bilas Khan, son of Miyan Tansen, after his father's death.
That gave birth to this raga, and that Tansen's corpse moved one hand in approval of the new melody.