The Title Track and "The Supermen" are more orthodox examples of this trope.
"The Thing That Should Not Be" by Metallica is about such a creature and, quite obviously, is directly inspired by Lovecraft.
Also, "The Call Of Ktulu" (though it's an instrumental).
In Guitar Hero, Metallica's "Metallifacts" video for "The Thing That Should Not Be"; it also lists "All Nightmare Long" and "Ride The Lightning" as being inspired by H.P. Lovecraft's stories; specifically, "Shadow Over Innsmouth."
Blue Oyster Cult is very fond of including these. Their Imaginos album plays with it and the song E.T.I. (Extra Terrestrial Intelligence) offers these lyrics:
"I'm in fairy rings and tower beds/Don't report this three men said/Books by the blameless and by the dead/King in Yellow, queen in red... all praise/he's found the awful truth..."
The song Harvest Moon from their Heaven Forbid album features the following lyrics towards the end:
"I sense the darkness clearer/I feel a presence here/A change in the weather/I feel some evil here/I hear some frightful noises/I don't go out at night/Since Bobrow's youngest daughter/Disappeared from sight/I know they'll find her some day/They find them all that way..."
And in the song Sub Human, which appears in two versions (Secret Treaties and Imaginos):
"Oyster boys are/swimmin' for me/Save me from the/Death-black creatures..."
Nile writes about a few of these in their songs, sometimes referring to Egyptian deities and sometimes to Lovecraftian ones. The best example in their music is from "Von Unaussprechlichen Kulten", which is essentially a retelling of the story "Dagon" by Lovecraft, only this time involving serpent folk: "I hath dreamed bleak and grim dessolate visions of the pre-human serpent volk, and communed with long dead reptiles, silently watching throught the ages, in cold curious apathy, the unending sorrows and sufferings of an abysmal human kind."
The title track of Vektor's Black Future album seems to deal with an entity akin to this that also acts as The Virus and a Lotus-Eater Machine.
Pretty much everything by Internal Suffering either has something to do with one of these or messing around with magic related to them.
Being heavily inspired by Event Horizon, the storyline for Luca Turilli's Prophet of the Last Eclipse seemed to be going in this direction. The demons could likely qualify, as well as anyone touched by The Black Portal, though Dark Is Not Evil in the latter case. And if Riders of the Astral Fire is any indication, those are only a warm up for what's coming. Unfortunately, the sequels never happened.
In the Talking Heads song "Air", the very air itself is an Eldritch Abomination. Wow. "Air hit me in the face. I run faster and faster... into the air. And I say to myself, 'What is happening to my skin? Where is that protection that I needed?' Air can hurt you too, air can hurt you too. Some people say not to worry 'bout the air. Some people never had experience with air. It can break your heart..."
"Azathoth" by Proto-Prog group Arzachel, which is about the Lovecraftian god of the same name.
The titular universe-devouring force in "The Great Annihilator" by Swans probably qualifies.
The Fame Monster from Lady Gaga's concert performance of Paparazzi. Resembling a cross between an octopus and an anglerfish, it's terror personified, complete with tentacles. Still, not too much trouble if you've got a sparky bra on hand. Watch here.
Electronic Ambient group The Orb has a song called "A Huge Ever Growing Pulsating Brain That Rules From the Center of the Ultraworld". Apart from being a Long Title it's a good description of what an Eldritch Abomination isnote It's also a perfectly sensible and accurate description of an episode of Blake's 7. The song itself is pretty soothing and strange in its own way.
The giant disembodied hand referred to as The Presence from the Nine Inch Nails album Year Zero. Along with some other unknown entities, it arrives to "wipe this place clean" if humans don't stop killing the planet.
The Black Dahliah Murder's song "Thy Horror Cosmic" is about one.