I used to wonder about that, and would be interested if anyone has the answer
Looks as if it's just a hangover from when spelling wasn't as settled as it is now, and you could spell it either way. It does happen with terms which are largely used in some single industrical context.
1) . Feigned, false, or counterfeit; sham; = faint adj. 1. Now rare.
a1400 (1325) Cursor Mundi (Trin. Cambr.) l. 19535 Şerfore toke he bapteme feynt [Vesp. faint].
c1400 Rom. Rose 433 She gan..To make many a feynt praiere To God.
c1698 J. Locke Thoughts on Conduct of Understanding §33 Dressed up into any faint appearance of it.
1702 London Gaz. No. 3835/2, The Major..made a feint Retreat.
1704 London Gaz. No. 3986/2, Amusing the French with..feint Marches.
1854 Thackeray Newcomes (1855) II. ix. 90 We wear feint smiles over our tears and deceive our children.
2. In commercial use, the usual spelling of faint adj. 5c; freq. quasi-adv.
1859 Stationers' Hand-bk. (ed. 2) 72 Feint only, the term for a book having merely feint blue lines across the page from left to right.
1895 Army & Navy Co-op. Soc. Price List 15 Sept. 525 Foolscap PaperRuled with Money Columns and Feint Lines.
1930 Publishers' Circ. 13 Sept. 321/2 The actual book itself should be of foolscap size, ruled feint.
Thanks Emma - much appreciated. Interesting.
I hope you're fully well again now? X
The two meanings (of feint and faint) do seem to have the same etymology, strange as that seems. My electronic Concise OED says they come from an Old French verb which seems to be spelled either "feindre" or "faindre", which apparently meant "to feign".
The two alternative French spellings seem odd, to me, as they would be pronounced differently in modern French. I've no idea whether either variant is still in use.
Fie on any faint-hearted fool who would fain feign a feint!