Similar to coinín in Irish
I'm reminded of proto-Germanic kuningaz "king" (compare Finnish/Estonian kuningas which was borrowed from Germanic), and the similarity of Slovak králik "rabbit" to kráľ "king" - perhaps it means originally "little king".
I wonder whether the Welsh word is also somehow related to "king" or whether the resemblance is pure coincidence?
Doubt it, as (like English "coney"/"cony") coinín and cwningen come from Latin cuniculus, ultimately from Basque/Iberian/Celtiberian. https://en.wiktionary.org/wiki/cony#English
Wiktionary even suggests that cwningen is a very old loan from English. https://en.wiktionary.org/wiki/cwning#Welsh
Old/Middle English "coni" and "cunning" have always been distinct words from "king"~"cyning", but it does seem that the Slavic words for "rabbit" could be a calque from a (deliberate?) misidentification of the words in German. https://en.wiktionary.org/wiki/kr%C3%B3lik#Polish
There is also the spanish "conejo" and dutch "konijn", that's quite strange to find it also in romance languages...