One view of the name Skelly from one interesting book…
…as previously reported, and popularly, though incorrectly professed, by those companies that sell genealogical coats-of-arms, - the origin of skelly is probably not the Sc. gaelic sgeallag which means 'Wild mustard, Sinapis arventis'. It probably is not Skelloch which means 'To shriek, yell or resound shrilly'. Although, it is possible that our Skelly family name originated from one of those sources or maybe from Skell' which means 'A shell'; or from skellach which is a 'small bell'. I certainly hope that the name doesn't originate from Skellum which means 'A rascal, scamp, scoundrel, villain'. What seems most likely to me is the following :
SKELLIED, a. Sc. [Cf. Skelly v.] Squint-eyed; squinting.
1821. Hogg, Jocobite Relics Ser. II. 40, "There's gentle John, and Jock the slorp, And skellied Jock, and bellied Jock." 1870. J.K. Hunter, Studies Char. 257, "Willie had a real earnest servant-lass, who was skellied in one eye." (OED : page 131)
SKELLY, Sc. rare, [Of doubtful origin: cf. Irish 'sceilg' reef, rock.] A rock, a skerry.
1513 Douglas, Aeneid v. iv 89, "On the sharp skellies, to hir wanhap, (she) Smat with sic fard the airis in flenderis lap." 1819 W. Tennant, Papistry Storm'd (1827) 102, "The storm that dang him frae the deep, Upon our tangly skellies steep." 1839 T. Chalmers in Hanna Mem. (1852) IV. v 81, "The rocks not high; but having more the character of skellies projecting into the sea." (OED : page 132)
SKELLY (ske'li), v. Sc. and north. dial. Also scalie. [ad. ON. *skjelga (cf. Norw. skjegla) refl. skjelgask, f. skjalgr (Msw. skalgh, skielgh, Norw. skjegl), adj.;=OHG. scelah, sceleh (MHG. schelch), OE. sceolh, squint, skew.] intr To squint.
Cf. Skellied a.; the noun and adj. skelly, and the comb kelly-eyed, are also in direct use. 1776. Herd, Scottish Songs (ed2) II 171, "Sae proud was he o' his Maggie, Tho' she did baith scalie and squint." 1816 Scott, Old Mort, iv "It is the very man, said Bothwell - 'skellies fearfully with one eye?'" 1829 - in dialect glossaries (Northumb., Burh., Cumb., Yks.) (OED : page 132)
The Oxford English Dictionary. Volume IX, S-Soldo. Oxford at the Clarendon Press. (Vivian Ridler) Oxford University Press, Amen House, London, GB. 1933, reprinted 1961.