Where does this odd expression come from, and what is it a chip of? A chip off the ole block, perhaps? What a funny cliche!
Phrases.org’s take on it. I do not know how reliable this is
OK, I’ll buy that…sounds logical enough!
Michael Quinion on World Wide Words gives the same explanation, but with this additional quotation:
What that suggests to me is that it was an attempted means to beat someone up while avoiding a charge of assault. If your opponent knocked the chip from your shoulder, that would be considered the first blow, to which you could “legitimately” retaliate.
I doubt it would have stood up in a court of law, but it seems plausible as a bit of “street lawyering”.
Or to sell a battery 
It seems to me I’ve heard a similar explanation for the phrase “cross the line”. You draw a line in the dirt, and if the other person “crosses the line”, it means you are going to fight.
I wonder if there’s any truth to that.
Yeah, there is. Kids did that now and then when I was young, in the dinosaur age, and it almost always resulted in a fight.
As to the chip on the shoulder, I never saw that happen, but when I was young it was a very common phrase in books and magazine stories. I think it was common in rural areas a generation or so before mine (which is nearly extinct now).
This early cite from the OED gives the original idea:
Gosh, that looks familiar. I could almost be persuaded I’d seen it before, just recently.
Twice.
I dare ya to quote the OED again!