There’s a Mount Shasta in Northern California. My parents and I have been there some years ago. 

I remember working at a garden center many years ago.  There were Shasta daisies flowers and seeds.  I don’t know if the flower has anything to do with the character in the Narnia books.

I've just looked it up, and the Shasta daisy was first cultivated in the US in 1890 (it's a cross between several species of daisy) and named after Mount Shasta in California, because of its snow-white petals. I don't know if it was popular in Britain in Lewis's time, but given that he doesn't seem to have had much interest in gardening — or in American mountains for that matter — I would guess that's not where he got the character's name from.

I really think it's most likely coincidental and he just invented the name "Shasta" himself as something that sounded vaguely foreign and exotic, similar to most of his other names from Calormen (Arsheesh, Anradin, Aravis, Ahoshta, Lasaraleen, Rabadash and so on). I reckon he had a good knack for coining names that somehow sound right for each particular character, even when those names don't have any literal meaning or obvious derivation. 

Posted by: @icarus

Posted by: @wren

Lewis pulled a lot of generic Middle Eastern tropes for his design of Calormen, I wouldn't be surprised if it might have been an influencer.

Wikipedia here says that Shasta is the name of a Hindu god, so that would certainly make a lot of sense, in terms of coming from that same general pool of inspiration sources:

https://en.m.wikipedia.org/wiki/Shasta_(deity)

That derivation wouldn't surprise me at all, when I think Lewis' tropes aren't strictly speaking, Middle Eastern or Persian, but also cross Asia Minor from Istanbul to Central Asia, & beyond, into the Indian subcontinent, where we have not only Tashkent, once capital of Uzbekistan, but also of archaeological discoveries involving Peter Frankopan, an Oxford University scholar, whose 2015 The Silk Roads: a new history of the world, suggest that humanity spread across Eurasia & elsewhere in the world, lured by trade as much as by displacement by warfare, conquest, disease, and much else. And though such trade involved furs, gold, silver, spices and much more, it also included slaves, as was also the case for the Ottoman Empire, which also spread into Europe, besieging Vienna, as late as July 14th to 12th September 1683. It seems that "the battle is noted for including the largest known cavalry charge in history".

I read online this statement:  Recently, archaeologists unearthed two exceptional lost cities in the Pamir mountains of eastern Uzbekistan, known as Tugunbulak and Tashbulak. Wikipedia informs me that the term, Tarkhan, if not Lewis' terminology of Tarkhaan, was used among the Sogdian, Saka, Hephthalite, Turkic, and proto-Mongol peoples of Central Asia and by other Eurasian nomads. It was a high rank in the army of Timur. Tarkhans commanded military contingents (roughly of regimental size under the Turkic Khazars) and were, roughly speaking, generals. They could also be assigned as military governors of conquered regions.

Precisely how Calormen's empire-building was organised, with its capital at Tashbaan.