It depends. Would I also wake up in that time period if I choose a historical figure? If so, I have a few of them. I'm fascinated by several periods of history, and would love to go live some time in each of these. Possibly my top three:
Siddhartha Guatama - The most famous Buddha, born in the sixth century B.C. in Nepal. I would love to walk a day in his shoes, both before and after his Enlightenment.
Masamune - Japanese swordmaker from 1300 AD. Not just for his prowess at swordmaking, but just for the insight of living in Japan at this time.
Gaius Julius Caesar - At the height of the Roman Empire. I think I'd try to spend the day sightseeing!
I could go on for pages, but I'll leave it at three for now.