“…Manwë is dearest to Illuvatar and understands most clearly his purposes. He was appointed to be … the first of all Kings: lord of the realm of Arda and ruler or all that dwell therein. In Arda his delight is in the winds and the clouds … Sulimo he is surnamed, Lord of the Breath of Arda.” — Silmarillion, Valaquenta
Not mentioned in this particular passage, but Manwë has a special affinity for birds, especially eagles, presumably due to his delight in the winds. Thus the winged circlet (I also like the hearkening to the winged helmets of Gondor). I maintain that this affinity for birds is the reason Gandalf has such a good relationship with Gwaihir the Eagle — before leaving Valinor to become Istari in Middle Earth, Gandalf was Olorin, one of the Maiar of Manwë.
All of the Valar took the guise of what they thought the Children of Illuvatar would look like when they finally came. Manwë has the overall look of a Noldorin Elf with dark hair and light eyes, which are both also reminiscent of the sky (this becomes more important when we later talk about Varda, his Queen).
I chose a dark stone in his circlet for a similar reason as the dark hair. Also because there seems to be an ongoing elemental theme with the other Valar, and with Ulmo being water I thought black was a better air color for Manwë than blue.