Radhanath Swami Maharaj describes that when Lord Caitanya appeared, the droughts ended. Rains and flowers of different seasons began to appear simultaneously. Considering all these auspicious signs, Nilamabar Chakravarti Thakur explained that the child was the nourisher & the maintainer of the entire creation. Hence He should be named Vishwambhar.
Saci mata was thinking that since her child was born under a neem tree, He should be named as Nimai. She thought that the neem tree is very effective in driving away ghosts or evil spirits. The Lord is all powerful & the master of the entire creation and no can cause Him any harm. However His yogamaya (internal potency) covers Him so that His pure devotees do not recognize Him as the Supreme Lord. The Lord loves to have sweet exchanges with His devotees that are beyond the level of just awe & reverence towards Him.
The ladies in the neighborhood named Him as Gaurhari because he looked very beautiful like Lord Hari with a golden complexion.
Jagannath Misra named him as Gauranga since every limb of His body was very delicate & had golden complexion.
– Radhanath Swami