The bandana has been used for a really long time for a lot of good uses, and its potential applications in the wilderness on backpack and camping trips is only limited by your imagination. Some immediately practical ones include:
Lightweight wash cloth - dries really quick with just a little wind or sun
Head cover - keep that scalp and forehead from getting red and shiny
Filter - attach to your water filter input line to keep the larger gunk out
Cooking - instant oven mitt
Wound care - wrap up a serious cut to protect it until you exit your trip
Drying - use it like a towel
Don't get lost - one of your choices is a bandana featuring topo maps for popular hiking areas
There's more of course, but you get the idea - a handy tool to add to your rental outdoor kit. These are good sized, flexible use cotton fabric at approximately 20" x 20". The topo versions are about $5 more than the standard. The colors on the standards will vary; either red or blue as available.