What's the origin of that pronunciation? I consulted with a Latin teacher at one of the local Jesuit high schools (he's one of the old school teachers; he's 66), and he said that though he knows of that pronunication, he prefers to pronounce "mihi" "mee-hee."
It SINGS a lot better than either mee-hee or mee-kee.
Honestly, I don't know...although logically, since the "h" is technically supposed to be silent, there would need to be some way separate the two "i" vowels with a stronger sound. Otherwise, one would say "mi-i", which would sound like a poorly elongated pronunciation of the vocative "mi".
Perhaps Solemnes made a judgment call when they published the Liber based on a custom in France. Just my thoughts.
It is the pronunciation from the Diocese of Rome. Our choirmaster is studying at Julliard and has recently checked out some books on Gregorian chant (which he said hadn't been checked out in 30-40 years, figures). He made copies of a pronunciation guide from one of them. When I get home I will see if I can tell which book it is from.
So, if you happen to assist at the indult Mass in Jersey City, you will hear the choir sing mihi as "mee-kee." It seemed a little strange to me at first, but I'm still learning.