Rumi sits next to 4 phonetic neighbors — Lumi, Remy, and Mimi are the closest matches by sound. Most share Rumi's Japanese cluster, so the rhythm and vowel shape carry across the list.
You searched for names like
"lapis lazuli beauty"
"brave, valiant (from Ry- element)"
"free woman"
"to know, perceive"
"sun, radiant one"
"enclosed valley"
"protector of men"
"free woman, strong woman"
"compatriots, fellow countrymen"
"a history, a recounting (archaic)"
"burnt woodland clearing"
"devoted to the god of wine and festivity"
"God is gracious"
"sequence of musical notes"
"divine, of the gods/spirits"
"illustrious, brilliant"
"bright, shining light"
"beautiful"
"dark, black-haired"
"slender"
Similar sound: These names share phonetic patterns with Rumi — similar vowel sounds, consonant clusters, or rhythm.
Shared roots: Several names on this list have Japanese 瑠美 origins, giving them the same cultural heritage as Rumi.
Rhythm: Rumi is a 2-syllable name. Names with the same syllable count tend to feel interchangeable when you're narrowing your list.