No.
Oh God, no.
No, no, no.
Nein. Nyet. Non. Huh-uh. Negative.
I'll qualify that. Extra credit that's built into the syllabus from day one, available to all students equally and in advance, can be defensible. I'd worry if it counted for very much -- a course grade should ultimately reflect performance on the core of the course, rather than the periphery -- but I can see an argument for moving a B to a B-plus in an art history class if the student does (and documents) some museum trips, say.
But this is the time of year when the kid who has been slacking or failing shows up, filled with sudden enthusiasm, begging for extra credit to make up for the work he either didn't do or did badly earlier.
From an administrator's perspective, this is a lawsuit waiting to happen. Don't do it!
Imagine: Johnny and Suzy are both on the cusp of failing. Johnny shows up and asks for extra credit assignments, and the prof. gives them. Suzy doesn't, believing that the grading system outlined on the syllabus is to be taken literally. Johnny passes with the extra points, and Suzy fails. Suzy finds out next semester that Johnny had an option she didn't have. She files suit, claiming disparate treatment. Your defense is...what, exactly?
Assume that Suzy has a couple of the magical 'protected class' memberships, too. Now we're talking civil rights lawsuits, with adverse publicity, multiplied damages, political overtones, etc.
Yuck, yuck, yuck.
Or: Johnny writes an extra credit paper, which turns out to be plagiarized. Can you fail him for cheating on an assignment that wasn't supposed to be part of the course grade in the first place? (If not, then doesn't Johnny have every incentive in the world to take a shot, gambling that he won't get caught?) Or maybe it isn't plagiarized, but it's a real steamer. Do you ignore it on the grounds that it's awful, or do you give something in recognition that it was extra work (even if the extra work stunk)?
I worry, too, about the cumulative effect of students encountering multiple extra-credit bailouts over the years. If students start to expect end-of-semester freebies to bail out three months of slacking, what, exactly, are we teaching them? Sometimes I think "suck it up" is one of the most valuable lessons we can teach. It's certainly an important life skill, and one that comes in handy at entry-level jobs. A kid who hasn't learned to suck it up is in for a rude shock when he gets to his first real job.
(I have a similar worry about makeup exams. A wise erstwhile colleague once shared her secret for getting around makeup exams: she'd give, say, four tests in her class, and count the best three. The students either stepped up or dropped the class. It struck me as brilliant, and I used it in my own classes to wonderful effect. Not having to distinguish 'excused' from 'unexcused' absences meant that I had to stop playing 'lie detector' when students told me stories about their lives. Grades reflected actual performance, rather than creative whining. Students either got their drama under control, or dropped the class and tried again when they were ready.)
Even-handedness can sometimes seem cold, and it can require saying 'no' when it would be easier not to. But the costs of ad hoc special favors are just too high to sustain. Fight the temptation!