From Merriam-Webster Collegiate Dictionary Main Entry: tinker's damn Variant(s): also tinker's dam /-'dam/ Function: noun Etymology: probably from the tinkers' reputation for blasphemy Date: 1839 : a minimum amount or degree (as of care)
From Microsoft Bookshelf tinker's damn tinker's damn also tinker's dam (tîng´kerz dàm)noun Slang. The smallest degree or amount: property that is not worth a tinker's damn.
[Probably from the reputation of tinkers for cursing.]
Excerpted from The American Heritage® Dictionary of the English Language, Third Edition © 1996 by Houghton Mifflin Company. Electronic version licensed from INSO Corporation; further reproduction and distribution in accordance with the Copyright Law of the United States. All rights reserved.
The etymology is foggy. Apparently the wordsmiths don't give a Tinker's Damn. Either variant is commonly used and as such is acceptable. Most likely the word dam meaning a barrier morphed into damn used as a curse. Anyone using the expression uses it as a term or derision and not as a technique of mending pots.