Innse Cait or Innse Catt is the name used for Shetland in ancient Irish annals. This seemingly is the closest thing we have to the ancient Pictish name for the islands. "Innse" is the plural of "Innis", which means an island in Gaelic; it is cognate with the Latin "insula", and also "Ynys" which is the Welsh language term. The Picts appear to have spoken a form of P-Celtic (i.e. related to Welsh, Breton, Cornish and Cumbrian) rather than Q-Celtic or Goidelic, but the term "Innse Cait" would have been rendered fairly similarly in both forms of Celtic.
"Cait" is the more interesting part of the name, for us. It means "of the cat". Cats are not indigenous to Shetland, so presumably this is a reference to the Catti tribe, who could be found in Caithness (Norse: Katanes) and Sutherland (known in Gaelic as Cataibh).
Neighbouring island groups
The neighbouring Orkney Islands were known as "Innse Orc". This is cognate with the Latin "Orcades", Norse "Orkneyjar" and English "Orkney". Orc means a pig or a boar, and may have been a totem of the tribe. It could also refer to porpoises or whales, which are sometimes known as seapigs in the Celtic languages.
It has been proposed by A. W. Brogger, that the Orkneys, Shetland and Faroe Islands all have names which represent Nordicised versions of Celtic originals, since Celtic language speakers such as the Picts were there first, and Papar/Culdees were involved in their Christianisation before the Norse invaded.
Innse Cait and "Shetland"
Hjaltland (sometimes seen as Hjaltiland) is the Icelandic name from the earliest period. however in the Faroe Islands and West Norway, Shetland was known as Hetland, which seems a lot closer to the Innse Cait, and may well derive from it. Hjaltland may just represent metathesis, i.e. the "t" and the "l" of Hetland being swapped around.
The modern Gaelic for Shetland is Sealltainn (pronounced "Shal-tin"), obviously a borrowing from English/Lowland Scots "Shetland", "Shaetlan" and "Sheltie". Otherwise, there appear to be little or no other old Gaelic names for places in Shetland, apart from Lerwick. Lerwick is known as "Learaig/Leuraig" sometimes, but this is again an English loanword, presumably brought back by Highland fishermen.