This name combines 来 (tai, rai, kita.su, ku.ru) meaning "become, cause, come, due, next", 來 (tai, rai, kita.su, ku.ru) which has the same meaning as the first one or 頼 (rai, tano.mu, tano.moshii, tayo.ru) meaning "request, trust" with 夢 (bou, mu, yume) meaning "dream, illusion, vision."
Raimu, as a name, is derived from a loanword which is a phonetic pronunciation of the English word "lime."