Madonna scholars (yes, they exist) have suggested that "my name" is a pun on the religious origin of Madonna's name.
Madonna, who was born Madonna Louise Veronica Ciccone, had an Italian-American father and a traditional Roman Catholic upbringing. In Roman Catholicism, the Blessed Virgin Mary is a highly revered religious figure. Over the years the Blessed Virgin Mary has had several titles, perhaps the best known of which is the Madonna (meaning meaning "my lady" in Italian). The figuring of "my name" as a pun comes from the idea that to "call [Madonna's] name" is to call the Virgin Mary's name.
"In The Midnight Hour" is a '60s R&B classic by Wilson Pickett and Steve Cropper. It's about sex.
The lyric in "Like A Prayer"—"In the midnight hour I can feel your power"—strongly echoes Wilson Pickett's verse: "I'm gonna wait 'til the midnight hour / That's when my love comes tumbling down." Another notable song to mention the midnight hour is of course the 1984 hit "Rebel Yell" by Billy Idol. The chorus of Idol's song begins, "In the midnight hour / She cried more, more, more." Clearly, the midnight hour is a mischievous one. By working it right into the song's chorus, Madonna creates the central question of "Like A Prayer": Is this supposed to be about God, or about sex? (Or both?)
As much as she has fought with the Vatican, Madonna has always been religious, growing up Catholic and remaining so until her recent conversion to Kabbalah.
Madonna has been vocal about her religious beliefs throughout the years. After the release of Like A Prayer she told Interview magazine that she broke from the church early in her life. She asked too many questions, like "But why can't I love God the same way if I have pants on?" when she became annoyed that she couldn't wear pants to school or church, and, eventually, "Why do I have to go to church to pray? Why can't I take the basic principles of this religion—principles like being good and doing unto others as you would have them do unto you and live with them? Why do I have to go to confession to confess my sins? Why can't I just tell God directly?" But Catholicism still shaped her worldview later in life. She said in 1989 that "I have a great sense of guilt and sin from Catholicism that has definitely permeated my everyday life, whether I want it to or not. And when I do something wrong, or that I think is wrong, if I don't let someone know I've wronged, I'm always afraid I'm going to be punished. I don't rest easy with myself. And that's something you're raised to believe as a Catholic. Everyone's a sinner in Catholicism, and you must constantly be asking God to cleanse your soul and begging Him for forgiveness."
Muses have a long, rich history in the Western world as symbols of inspiration and knowledge.
The idea of the muses—three to nine goddesses (depending on your sources) that inspire artists—dates back to very early Greek mythology. The conventional myth of the muses in today's world holds that there are nine muses, each associated with a different poetic art. Madonna might be interested in Erato, Euterpe, and Terpsichore, which are identified with lyric poetry, music, and dance, respectively. More generally in English, the term "muse" simply suggests "an inspiration," though that's a bit less interesting.