Recently, I read a book called "How TCS Built and Industry for India"
And I got to know that Lakmé is an Indian company which was started as a subsidiary of Tata Oil Mills Co.
After India attained independence, its economy was fragile. And like other industries, the Indian cosmetics market also relied heavily on international brands. The burgeoning middle and elite class was splurging on imported cosmetics and this had a direct effect on our foreign exchange reserves. 
Alarmed by this, the then Prime Minister of India, Jawahar Lal Nehru, approached industrialist Jehangir Ratanji Dadabhoy Tata in 1950 to start an indigenous cosmetic brand. 
The brand was named ‘Lakmé’, an English derivative of the goddess of wealth and beauty — Laxmi.  
Legends say that the aim behind having an exotic name was that Laxmi would not resonate with the idea of beauty, which is largely a western phenomenon.