'Mount Fang' is the Elvish rendition of Orthanc's name, ultimately from the roots orod 'mountain' and anca 'jaws'. The meaning 'cunning mind' comes from Old English orþanc, representing a word in the lexicon of the Rohirrim.
Tolkien himself is noncommittal on the name's origins. He says (in The Two Towers III 8, The Road to Isengard): '...the name of [Orthanc] had (by design or chance) a twofold meaning...'. It is possible, though, to speculate further on the beginnings of the name.
There are fairly good linguistic grounds for presuming that the 'Rohirric' meaning is the original, since orþanc is a real Old English word (actually a philosophical term, but meaning more or less 'cunning mind'). This name must have been given to the tower by the Rohirrim who lived in the lands around it, and would only make sense after Saruman's arrival (since his must have been the 'cunning mind' to which it referred). We can therefore fairly safely date the name 'Orthanc' as emerging in Rohan sometime after III 2759, the year Saruman took up residence there. That the name is also an approximate Elvish translation of 'Mount Fang' seems to be no more than a curious coincidence.
The tower itself was much older than the Rohirrim: it had stood for more than 2,500 years before the land of Rohan even existed. Its builders, the early Gondorians, would almost certainly have given the tower an Elvish name, but that name is no longer known. The name they chose for the whole complex of Isengard, where Orthanc stood, is recorded: Angrenost, 'the fortress of Angren'.