...as in, the pioneering late founder and abbot of Zen Mountain Monastery and the Mountain and Rivers Order? Well, I didn't either! But thanks to a comment left here, now I do and you do too. (Thanks, Josh L.) The band is on hiatus as of this past summer; here's a track called, appropriately enough, "Buddhist Priest," as well as live video from their last show.
Though defunct, you can hear more of them on their Bandcamp page.
As for why Daido Loori have named themselves that, I reached out to them to and see if they might share the story. First, drummer Charlie responded, letting me know that the band is not broken up, as I'd reported in an earlier version of this post:
"One of our vocalists, Isabel, studied under Daido Roshi if I'm not mistaken. I'll get her to get in touch with you, and explain a little bit more. Also, we're not broken up! We're on a hiatus, as our members go to school all across the Wast coast, but we're putting out a new track very soon, and hope to tour over the summer."
Next, I heard from Isabel:
"Hi! My mother was a student at Zen Mountain Monastery and a lot of time in my childhood was spent there, as well as my parents' wedding and my baptism. Daido Loori was a significant influence on my mother's spirituality as well as my own. When I was a small child, he taught me many values that have inspired me to this day. He was just as much of a friend as he was a teacher. I named the band Daido Loori in homage to him."
So there you have it. Thanks to Charlie and Isabel and the band as a whole.