Why is it called ירדן?
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via: Klein, Ernest, A Comprehensive Etymological Dictionary of the Hebrew Language for Readers of English, The University of Haifa, Carta, Jerusalem, p.264
While שיורד מדן is a rather neat derivation and mnemonic, the Dan part would seem to be unnecessary, since the nun ending seems morphological; also, we would be reusing the daled. Also, we would need to grapple with Biblical references to haYarden hazeh (to Yaakov), where it was surely not called Dan at that point, to spark the derivation. In sum, a nice derash, but I would favor Klein's related etymology.
The Sefer Mincha Belulah say from the Gemara in Bechoros because שיורד מדן it flows down from the North part of Israel, that belongs to Dan.