This sounds to me like a variation on a Spanish name - Spanish names which were incorporated early into the Native American population are often mistaken for genuine Native American names. If it were spelled Cojuita or even Cojita it would be pronounced approximately "co-WEE-ta" - I'm pretty sure? Or something similar enough to have been corrupted to that pronunciation?
"ita" is an extremely common ending to Spanish feminine names, which makes me pretty sure that the origin of Coyita would be Spanish. I don't know what name it would be based on, though - I'll have to keep looking.