No, the original form is Sloveni (Slavyani is a modernized Russian form, I think, or at least it became popular under imperial Russian influence a few centuries ago; of course, it didn’t quite spread everywhere, hence Slovakia and Slovenia; plus, the A form existed already in the medieval Greek transcription “sklavenoi”, despite the original Slavic O spelling of the time). Thus, Sloveni originates from “slovo”, i.e. words, speech. A common antipodean example is our name for the Germans – “nemtsi”, i.e. mute, unintelligible (what the Greeks originally referred to as “barbarian”, in much the same logic).