Using the prefix Thee rather than The began amongst Mexican-American gangs in the 1950s, I’d heard. The first band I know of to use it was Thee Midniters in the early 1960s, a hispanic group from East LA. The reason, in their case, was to differentiate themselves from another band that already existed called The Midnighters (the group led by Hank Ballard who did the original version of The Twist).
There was a British group called simply Thee in the mid-1960s but that’s totally unrelated.
There were a few other groups in the sixties who used Thee as well, for a similar reason. But it was the previously mentioned Billy Childish who resurrected the idea for one of his bands, not for the same reason, but in homage of those sixties bands that he’d admired. Other 1990s garage bands influenced by 1960s garage bands picked up on it too.