Luthien: Maia or Elf.

[Mithalwen]

I can't help wondering (and I know this is one of my pet hobby horses cf Music and Magic in Middle Earth ) if the fact that Luthien's power is manifested in song is important .

The world is created through music (simplification I know) and it is through song that Luthien charms both Morgoth and Mandos. Melian would have surely participated in the music of the Ainur and maybe something was passed on to Luthien. Finrod uses a "song of wizardry in his duel with Sauron"

Also bear in minds that Thingol was a mighty Elf Lord - one of the 3 Originals so to speak - to put it crudely the power of the elf lords seems to be diluted with each generation ( this is supported by the fact that elf parents put more of their own strength into their children and that consequently the larger families among them occured in the early years when they were at their peak). As a first generation descendant of an original elf lord, Luthien would be nigh on a par with Fingolfin and Feanor without the not inconsequential benefits of Melian as a mother.