Rayme sits next to 5 phonetic neighbors — Remy, Raimi, and Ramona are the closest matches by sound. Most share Rayme's Germanic (via Raymond)-origin cluster, so the rhythm and vowel shape carry across the list.
You searched for names like
"wise protector"
"stony ridge, rocky heights"
"descendant of Dobhailén"
"powerful ruler of ancestral lands"
"gifted ruler of the people"
"crest, summit, peak"
"Beautiful, lovely"
"God is gracious (through Juana/Juan root)"
"Eagle power; noble strength"
Similar sound: These names share phonetic patterns with Rayme — similar vowel sounds, consonant clusters, or rhythm.
Shared roots: Several names on this list have Germanic (via Raymond) origins, giving them the same cultural heritage as Rayme.
Rhythm: Rayme is a 2-syllable name. Names with the same syllable count tend to feel interchangeable when you're narrowing your list.