This isn't my area of expertise but I can't abide an unanswered question!

'The City of Brass' in One Thousand and One Nights describes the smoke that transforms into a genie as blue, so it does seem reasonable to carry the colour over:

When the fisherman broke open the bottle, witnesses said blue smoke poured forth and rose into the sky. A horrible voice cried out, ‘Repentance! Repentance! O Prophet of God!’ Suddenly the smoke coalesced into a huge and frightening person, high as a mountain, who just as quickly vanished into thin air.

That's perhaps where we get the contemporary blue genie. The idea of different coloured jinn seems widespread, but difficult to pin down. Robert Lebling writes in Legends of the Fire Spirits: Jinn and Genies from Arabia to Zanzibar:

According to some Islamic traditions, jinn are distinguished by colours. In Syria, the tradition holds that the king of the jinn is red in colour. Ernst Zbinden finds ‘a very charming correspondence between old and new perceptions’ in the notion that the most evil jinn are coloured red. But he was able to confirm this view only in North Africa. These general theories about jinn stem from ancient works on Arab magic, some of them handbooks or grimoires, mostly of Egyptian origin.

It's important not to think of the Islamic world as culturally homogenous - there's tremendous variety in its cultural traditions and the significance of the different colours seems regional.

According to Lebling, Malay folklore seems to be the most flamboyant with its colourful supernatural beings, but even these vary from location to location:

They believe that the jinn sprang from three mangrove-leaves: the green jinn from a leaf that soared into the sky, the black jinn from a leaf that fell at the gate of the forest, and the white jinn from a leaf that fell into the sea.

According to another tradition suggested by the wording of certain ancient magical incantations, the jinn were created from the soil of the mountain Mahameru, the Malay equivalent of Mount Olympus or, perhaps more appropriate in this context, Mount Qaf.

According to other incantation traditions, the jinn of the earth were born of afterbirth, or of the morning star. One magician’s account says that jinn sprang from the coconut monkey. Another asserts they were created from Sakti-muna, a great serpent: the king of the jinn from his breath; the white jinn from the whites of his eyes; the black, blue, green and yellow jinn from his irises and the jinn that lives in the lightning from his voice.

As an aside, blue was used in Turkish amulets to offer protection from jinn. This is where we get the word 'turquoise'. (EDIT: I got overexcited and combined two unrelated facts)

Hi There, Thanks for the fairly rare question(s). /u/JDHoare took a few words out fo my mouth for coloration so I'll tackle a few of the other aspects. Genies, as part of pre-Islamic mythology, are kind of like humans in that there is a bit of variety about them, they can be mercurial, and they are in many ways like humans, only "created" by God out of different materials (humans were clay, genies could be air or fire).

The gold armband as depicted in Western film is mostly part of the perceived "Arab" culture of the time, but there's no reason not to think that Genies themselves did not wear their wealth, so to speak, because many of the stories which involve Genies in mythology speak of their sometimes wealthy appearance. It is interesting to note that it seems to be inferred in many myths that Genies can change their appearance as well, to suit the trick or task they have in mind for the human (i.e. a prince of Genies will appear princely, gold and all). Genies, however, can be pretty mean, or sometimes downright evil (much like people can be) to the point of driving humans insane. For example, when Muhammad received his first visitation from the angel Gabriel, he ran home and hid under the covers of his bed until Khadija convinces him that Genies were not trying to drive him mad.

In the stories in One Thousand And One Nights, many of which are either borrowed from or reference India, the Genies may be borrowed from depictions of the multiplicity of Gods in the subcontinent as well and thus have a bluish (or red, or other color) associated with them, though not specifically their skin.

Suggested reading:

Inea Bushnaq's Arab Folktales

Khairat al Saleh's Fabled cities, princes, and Jinn from Arab Myths

All removed comments were unsourced speculations or offtopic suggestions.