The more likely root in my book would be 'sláva' - fame or glory

I think slov- 'word' and slav- 'glory' are connected: I seem to recall reading something recently (I suspect Lehmann's book on Indo-European) which made the claim. For the vowels, there's the ethnonyms Slovene and Slovak. For the semantic connexion, cf. Greek doxa 'opinion; glory', where the semantic shift is via a common sense 'reputation'.

Afterthought:

I might be confabulating at this point, but I even have an idea that the underlying Slavonic slew- 'speak' is Indo-European klew-, and therefore related to loud and ?listen and Latin cli-ent.