For cheap, look on ebay for a really old model – like a Garmin GPS III or something like that.
I have one. It is several years old. B/W screen. Anchient maps. No navigation guide. Still, it does two things well: Lat and Long. It would probably serve you well on your hike.
It has some nice features, too. You can offload data to a PC, or plug into a serial port on a laptop and superimpose the lat/long in real time on some mapping applications.
