Alaikumassalam wa rahmatullah,
Raamesh [pronounced as (R)andom + sm(A)rt + (ME)dal + (SH)ape], which can also be spelled as Raamish, is a Farsi word that means happiness, cheerfulness, pleasure, and melody, as in this line by the poet Fridausi:
پادشا بادی با رامش و آرامش دل
آشنا بادی با دولت و اقبال و جلال .
To be a king with raamesh (happiness) and araamesh (calmness) of heart
To be accustomed with statecraft, fortune, and splendor
The Farsi dictionaries, old and new, make no reference to any Hindu relation regarding this word. The Hindu name is derived from Rama + Eesha, references to Hindu gods. But the Farsi Raamish is derived from Pahlavi Raamishn which means "happiness". For this reason I don't see any relationship between the two words, and inshAllah you can happily keep your son's name as it is.