I like my strap stored in a nice clear zippered bag, like what you would get bed sheets, or cheap jumper cables in. i save every one of those bags i get just for storing stuff in my rig. they are reasonably water resistant (short of being thrown in the river), you can see what's inside, Another thing that is handy is military ammo cans, you can get many different sizes, you can paint them, you can drill holes in them to bolt them to brackets. and they are extremely water proof. I like to keep a small 12v air pump, tire plug kit, emergency kit, and emergency blanket in mine. and if i choose to, i can stencil on it what it is, if i wind up with more than one. also, these are excellent for tire chains. keeps the mud and rust out of your rig, then after you get home, just remember to rinse them off, let them dry,and then hit them with a coat of paint or wd-40 before putting them back into the can.
I even suggested this to a friend who does off road racing... cut a hole in the end of an amo can, glue some plexi glass on it, make some padding inside with trash bags and sprayfoam to fit a small video camera inside it, then bolt the can to a mount on his roll cage and he could video his race without having to worry about the little tripod mount snapping out of the camera and having a bunch of trucks drive over it. it worked quite well