The short answer is that the author uses strain as a verb here, with the meaning plays or sounds.
There's a long history of the noun and verb forms borrowing from each other. And there are many many words in English that are both nouns and verbs. There is an obsolete verb meaning that refers to playing an instrument or singing a song. Here the verb is intransitive. EDIT: please see my note at the end of the answer, regarding the obsolete verb strain.
The noun strain can refer to
A strain is also a particular sound:
Strains of piano music drifted across the room.
Note the example sentence of Cambridge refers to the strains (sound) of piano music. Compare this to a definition in Oxford:
The sound of a piece of music.
So, simply put, the author uses the noun strain as a verb and applies this to the piano music (which is what Debussy's "Clare du Lune" is) that the blind Marie hears in the room over her. the
The author could have written
...and then her grandfather’s song, “Clair de Lune,” plays or sounds over her like a blue mist.
This 'over' could be strictly literal, since the gramophone is in the room (I think it's actually the attic or garret) over or above her grandfather’s bedroom.
'Over' could also be used metaphorically, similar to The music washed over me; here it is like a blue mist.
Why a blue mist? Debussy's "Clare du Lune" is a very evocative piano piece, one that is true art, in that it causes a lot of responses in most people who listen to it. Hear it at, among others, this recording and read some of the many among 9000+ comments about 'what the song evokes in or to me'. One thing the song can evoke is the image of a blue mist.
"Clare du Lune" means, in French, "moonlight", and that word alone can call up images of light through a dark blue sky; you simply have to add some mist to get some blue mist.
A search of the available text to the novel on Google Books shows us that this is the third or fourth mention of the song "Clare du Lune" in the novel.
Besides the connection with the song's meaning of "moonlight" with the book's title, it's also important to note that the author tries hard to use descriptive language, including verbs, when describing the strains (sound) of this evocative piano piece. Elsewhere in the novel he talks about the piece thusly:
a song that makes her think of leaves fluttering and of the hard ribbons of sand beneath her feet at low tide. The music slinks and rises and settles back to earth.
Note that mist might also be thought to slink and rise and settle back to earth.
Chords float past in transparent riffles.
...and then her grandfather’s song, “Clair de Lune,” strains over her like a blue mist.
the author's just using a noun as a verb to describe the sound that the piano makes 'over her'. Or he's using an obsolete sense of strain. But even this obsolete usage displays a close connection between the noun and verb forms.
As mentioned by Damkerng there is the obsolete meaning of strain as a verb to refer to singing or the playing of an instrument. But obsolete means "no longer used." So either the author decided to employ this obsolete usage or he used the modern noun in a creative he way as a verb. I imagine that 99% of contemporary native speakers are unfamiliar with this obsolete usage. (That's why it's marked as obsolete in the Oxford English Dictionary or OED.)
There are many many words in English that are both nouns and verbs, so it's not surprising that there is an obsolete version of strain to refer to the sounding/playing of an instrument, including the human voice. Also, the OED hints that even the obsolete meaning probably was influenced by the noun forms.
To me, strains as a verb would work much better with violin music, since (a) the strings of a violin produce actual physical strain or tension on the violin; and (b) the unique sound of violin music is often referred to as strains. I just don't see "strain" working very well with piano music, especially one that, to me, is pretty free of strain (in the sense of tension or something that puts a constraint on people). And to say that something strains over someone like a mist is, in my opinion, not the greatest simile.