Hays sits next to 3 phonetic neighbors — Hayes, Haze, and Maze are the closest matches by sound. Most share Hays's English-origin cluster, so the rhythm and vowel shape carry across the list.
You searched for names like
"hedged enclosure"
"playful one"
"messenger of God"
"blazing torch, flaming sword"
"new house"
"Yahweh saves, God delivers"
"from the clearing"
"reddish-brown color"
"leader, head, principal one"
"from the island of Cyprus"
"grace of God + home ruler"
"light, illumination"
"thunder and lightning (via Raiden)"
"homestead on the rock"
"handsome, well-born"
"victorious"
"herald, messenger"
"elder, presider"
"one who works at the latticed barrier"
"beautiful, handsome"
"from the River Clyde"
Similar sound: These names share phonetic patterns with Hays — similar vowel sounds, consonant clusters, or rhythm.
Shared roots: Several names on this list have Old English origins, giving them the same cultural heritage as Hays.
Rhythm: Hays is a 1-syllable name. Names with the same syllable count tend to feel interchangeable when you're narrowing your list.