The real etymology is unknown. This is very often the case with place names. Here are the options that are on the table, but it's overwhelmingly likely that none of them is right.

• One attempt has been made it to tie it to a verbal adjective spart- from speir- "to sow", meaning "sown" (as in the Spartoi, the "sown men" who sprang from the dragon's teeth at Thebes; i.e. nothing to do with Sparta), and at least one ancient source uses a wordplay with Sparta and spartē/sparton "a rope, cable" which comes from the same verb, but so far as anyone can tell that's just playing with a chance resemblance.
• A couple of ancient sources mention a cult to a legendary "Spartē", daughter of (the river) Eurotas and wife of Lakedaimōn, at Amyklai near Sparta, and one of them claims that the town was named after her.
• An early Argive source, Akousilaos (5th cent. BCE), reports that the first man on earth in Argive legend, Phorōneus, had a son Spartōn, who in turn had a son Mykēneus. Another source, perhaps based partly on Akousilaos, reports that Spartōn and Messēnōn were the sons of Phorōneus' brother, Phēgeus. A 2nd-cent. CE source tells us that the Spartans of that time rejected this story, but of course that's 700 years after Akousilaos, so who knows what people were thinking in the 5th cent. BCE.

We can rule out the second and third possibilities in any case, as eponymous legendary figures like these are nearly always post hoc rationalisations for existing place names: that is, "Spartē", "Spartōn", "Mykēneus", and "Messēnōn" were invented as just-so stories to explain the names Sparta, Mycenae, and Messene. The speir- "sowing" etymology isn't actually impossible, but there's exactly nothing in support of it, other than chance resemblance.

The etymology of Lakedaimōn, the name by which the city and the people were normally known ("Lakedaimōn" was the original name, attested in the Bronze Age; "Sparta" was a later arrival), is also speculative. As with Sparta, a few suggested etymologies exist, but none of them are at all compelling.

Nothing specific, really. The "state mythology" was that the legendary founder of the city of Sparta, Lakedaimōn, had a wife named Spártā and named the newly founded city for her. But Sparta in ancient times only referred to the city proper, anyway. The state was called Lakedaimōn, the inhabitants Lakedaimōnioi. Hence the Λ symbol commonly depicted on Spartan shields - the greek letter lambda, or L, for Lakedaimōn.