What does the expression “red-headed step child” mean?
A step child with red hair.
I’ll beat you like a red-headed step child.
I’ll beat you like a rented mule.
Both popular sayings in my circles 
A red-headed stepchild, according to various sources and conversations, is a stepchild – that is, a spouse’s child who is not your own – who has red hair, which physically reminds the (presumably not red-haired) step-parent of the relationship that his/her spouse had with someone else.
“I mean, sure, I’d like to forget that she married Sam instead o’ me, and since I ended up with her, I oughta be happy, but every time I look at that boy, I see Sam grinnin’, and I get so mad, I just wanna wipe that damn grin off his face.”
It essentially means the least favorite. I guess a stepchild would be the least favorite child and maybe a red-headed one would be even less desired (or possibly remind one that the child isn’t theirs). A search of the web turned up no definitive explanaiton for the origin of the term. As a redhead, I find the term irksome…
Thanks. It doesn’t make much sense to me either. I guess it means, like you said, easy to pick out and pick on. I guess the fact that read-heads don’t pop up without their being some redheadness in the family suggests that maybe the kid is “illegitimate.” My nephew is a very cute red-headed genius, so the label is “irksome” to me, too. 
It implies a bastard.
The Irish were once horribly discriminated against in America.
The implication is that the “stepchild”–i.e. a euphemisim for a bastard-- is not only unwelcome, but of an unwanted ethnic group. Hence the savage mistreatment.
We must be in the same circle because whenever I say the first I invariably follow it up with the other.
Red hair is stereotypically associated with wildness and rebelliousness.
Step-children are stereotypically associated with rebelliousness (as step-parents are stereotypically associated with withholding love).
Beyond the point that stereotypes are invidious and often wrong:
A red-headed step-child would have the combined traits to need the most “correction.”
Let’s stop dancing around the obvious people. Redheads are just plain more beatable. <ducks>
It reminds me of Annie
While I’m not sure it is entirely relevent, in Anne of Green Gables the heroine, an orphan adopted by an elderly farm couple who are brother and sister, feels self-conscious about her looks and observes that “there is nothing worse than red hair”. Evidently red hair was viewed as a flaw in some circles in late nineteenth century America, possibly on account of the hostility expressed towards Irish immigrants.
As previous posters have noted, the expression refers to a person who is treated as the odd man out, and less favored than others.
I recall too reading an article in Parade Magazine once which claimed that there was a “well known” stereotype that men dated blondes, married brunettes, and cheated with red haired women. I’ve never heard that before or since, but if you can’t trust Parade Magazine, what’s left?
As for the rented mule, IIRC, this used to be a favorite catchphrase of Fibber McGee.
I’ve only heard “she screamed at me like a red-headed stepchild.” Maybe that’s because my lead singer is red-headed.
Not that I’ve advanced the discussion any…