Beorn's name 'Carrock' is nowhere explained (and indeed conceivably it had no intended meaning whatsoever). However the element '-rock' points to at least a partial derivation from English, and the Old English carr also means 'rock', so perhaps this is simply a duplicative name combining Old English and modern English words for 'rock'. Variations on this word also appear in Celtic (it seems to have been borrowed from that language by the Anglo-Saxons) and it occurs occasionally in real place-names, perhaps most notably the Carracks, rocky islands off the coast of Cornwall.
An alternative derivation comes from Old Norse, which gives us carr meaning 'wetland, river', and thus 'Carrock' would mean simply 'river-rock', which would make sense given its location within the stream of Anduin.
Finally it should be mentioned that 'Carrock' occurs as a real place-name in northern England, an anglicisation of Celtic cairoc, 'fortress', but this seems unlikely as a source for Tolkien's choice of name.