Ermine

hoary - O.E. har "gray, venerable, old," the connecting notion being
gray hair, from P.Gmc. *khairaz, from PIE *koi-. Ger. retains the
word as a title of respect, in Herr.

ermine - c.1175, from O.Fr. hermine, both the animal and the fur,
apparently from a convergence of L. (mus) Armenius "Armenian
(mouse)," ermines being abundant in Asia Minor; and an unrelated
Gmc. word for "weasel" (cf. O.H.G. harmo "ermine, stoat, weasel,"
adj. harmin; O.Saxon harmo, O.E. hearma, etc.).

(Douglas Harper, ETYMOLOGY DICTIONARY [of English language])

Also Lithuanian s^erm-uo~, serm-uon-e:~l-is
Lv. sermulis

what is origin of that word?

As my knowledge reaches both germanic h and baltic s^ came from
proto-indo-european *kh. So it shows that root was *kherm-/khorm-.

Also lithuanian s^`irm-as/-is (grey, white, hoary), as i understand
it came from pie k(h)r.m-os. Could English hoary be related to this?

btw s^irmas is related to s^ermuo~ (white animal) because
analogically Old Prussian Gaylux (ermine) is derivetive from Gaylis
(white, grey).