mi is not 'beauty' but 'deity' ko is 'daughter', thus Miko means 'daughter of a deity', often translated with 'priestess'.
according to a very good online Japanese-English dictionary, haru means 'spring'. Harumi would mean something like 'god of spring'.
to be accurate: mi means 'three', 'seed, good result', 'body, oneself', not yet' and it is the sixth sign in the Chinese astrology (the Serpent). So, Harumi could also mean 'seed of the spring', as in 'born in the spring', like Haru.