1868 March, Mr. Thom. White's Little Sermon, in Putnam's Magazine, page 555: "No, Peter; women, now, are not worth a tinker's mill-dam—that's what I think." [...] Just what a "tinker's dam" is, I have no means of knowing; but I believe it to be something very worthless indeed.
↑ “tinker's damn” in Merriam-Webster Online Dictionary, Merriam-Webster Online.
↑ 1858 December, January Searle, Hyra, the gipsy Prophetess, chapter XXI, in Frank Leslie's New Family Magazine, volume 3, number 6, page 546: "[...] and I doesn't care a tinker's dam about him."
1 2 “tinker's damn” in Douglas Harper, Online Etymology Dictionary (2001).
Etymology
Merriam-Webster finds tinker's damn in print since 1839 and suggests that it derives from tinkers' reputation for swearing.[1] The spelling tinker's dam is attested since 1858,[2] and phrases.org.uk notes the disagreement over whether the term originated from tinkers' swearing or instead from their use of small, single-use dams to hold solder.[3] The latter explanation has been offered since 1877;[3][4] on the other hand, the phrase tinker's curse is attested since 1824 and the phrase worth a tinker's cuss is attested since 1854, for which reason Etymonline considers the "dam" derivation an "ingeniously elaborate but baseless" invention of later writers.[4]