While a spare set of clothing is a good idea, it's also worth mentioning that toiletries tend to go hand-in-hand with spare clothes. That's why I always keep a Dopp Kit (or toiletry kit) made up for myself. Whenever I pack spare clothes it gets tossed in the bag.
Having that kit means I have soap, a washcloth, shampoo, deodorant, shaving stuff, toothpaste/toothbrush/floss, hair brush, a little mirror, tweezers, band-aids, Tums/Advil/Pepto, and some other little miscellaneous items.
More often than not, I've found the Dopp kit to be more important than the spare clothes. I mean, there have been plenty of times where I've had to sleep in my clothes overnight (college memories anyone?), but it was never really a big deal as long as I could find somewhere to clean myself up a bit in the morning until I could get home.
I mean, my clothes might have been a bit dirty/wrinkled, but at least I looked and smelled kinda clean (even if I was suffering from a terrible hangover).
[On a side note, if you're looking to make up a cheap toiletry kit, check out your local dollar store. They usually sell all the basics and in small travel sizes, perfect for making up a little kit.]