Writing in The Guardian (8 July 2011) Amit Chaudhuri's "Rereading Rabindranth Tagore" explains the significance of "Rabindranath"
"Tagore's first name sounded like gobbledegook to Larkin's ears, and Dickens, who met Tagore's grandfather Dwarkanath in London in 1842, had this to say of that name: "I have spelt it backwards, but it makes no less tremendous nonsense that way." But there's a narrative behind the names. "Dwarkanath" means "lord of Dwarka" – Dwarka is Krishna's home; it's another name for Krishna, and is a properly Hindu name. Tagore's father's name, Debendranath, means "lord of the gods", and has a clear religious connotation. "Rabindranath" means "lord of the sun"; it announces a shift from the invocation of the gods in Bengali naming toward names that suggest or contain light or radiance. Debendranath, a prime mover of the unitarian Brahmo Samaj, is, in naming his son (indeed all his sons), moving away from the old, populous Hindu universe to a sphere of immanent illumination: the world of the so-called Bengali "enlightenment".