Based on this claim:
Rami is a form of the Hebrew and Indian name, Ram.
Regarding Ram, the same site says:
From Hebrew roots, its meaning is 'exalted, supreme'. The name was
borne by the second son of Hezron, a descendant of Judah in the Old
Testament of the Bible.
Sure enough, Ram is mentioned in Ruth 4:19:
וְחֶצְרוֹן֙ הוֹלִ֣יד אֶת־רָ֔ם וְרָ֖ם הוֹלִ֥יד אֶת־עַמִּֽינָדָֽב׃
and Hezron begot Ram, and Ram begot Amminadab
It's possible that "Rami" was similar at that time to adding the "i" vowel sound to names then, as many do today such as Shmuli. Avrami and Chani, etc.
I am surmising that the Egyptian name for Paroah, Ramses, may have emanated from the same root name. Similar idea may apply to the famous "Romulans" from one of the old Star Trek episodes.