For a lyricist--a diverse vocabulary, an ear for rhythm and rhyme, an associative mind, a vivid imagination, and a love of resolving puzzles.

For a composer--12 notes, a smattering of sharps, flats, naturals and accidentals, an infinite arrangement and intuitive understanding of chord structures, an ear for rhythm, an associative mind, a vivid imagination, and a love of resolving puzzles.

For both--some attributes described by some other writers of note:
As Jerry Rasmussen noted above, a keen eye for observation is essential.
Nathaniel Hawthorne said you need "ice in the blood"
Edgar Allan Poe said an effective work needs "continuity of effort, unity of effect"
Muriel Rukeyser advised "keep your word hoard dry"
Alexander Pope said "the sound must seem an echo of the sense"
Robert Frost said you need the ability to make the "commonplace unusual and the unusual, commonplace"

I agree that just before waking and just after falling asleep are excellent times to harvest lyrics and melodies from the subconscious. Sometimes they're fully formed and just need tweaking, but more often they come as fragments and shards that (as T.S.Eliot said) slip and slide and decay with imprecision. (I think that's from Ash Wednesday).
Consequently, they need lots of labor and time.

Somebody else said that writing is easy--just sit down and open a vein.

It is not necessary, in my opinion, for *good* songwriters to have any performance persona at all. If they do, well and good. If not, it doesn't matter--they won't be performing the song a hundred years later. The song is the thing that must transcend time, place, and performer.

Just my $0.02, fwiw...

V