Most compact digital cameras come with a lousy wrist strap. I say “lousy” because they often lack a slider to fit them to your wrist (which makes them more prone to fall off). And in any case you can’t put them around your neck, which is often a better choice if you need both of your hands when you’re out and about.
That’s why I’ve replaced those straps on my compact cameras with this Hakuba neck strap. I first found one about six years ago when I nearly smashed my Nikon CoolPix 880 on rocks while on a hike with my newly wedded wife in the Virgin Islands.
The strap is thick nylon which feels sturdy around your neck. Wearing the camera around your neck gives you both hands free to hold onto rocks or your spouse (hey, we were newlyweds then!). It also has a slider so that you can turn it into a wrist strap (albeit a long one) when you don’t want to look like a dork with a camera around your neck, and it has a quick release for when you need to quickly detach it from the strap.
There are other straps that might be OK, but I like the durability of the Hakuba strap.