I have a 120cm nylon runner that I want to use to tether me to an anchor bolt while cleaning the anchor. The runner has a small fray (pictures below). Is it safe to use?
Retire that sling. Use it for some handicraft work at home or similar.
Firstly: If in doubt about a piece of gear, don't entrust your live to it.
Doing a more detailed analysis without seeing it in real live is hard, but here are two arguments apart from the doubt one (which is enough) that come to mind:
It looks like there is a discoloration right on the left of the fray. That's a sign of UV influence, which reduced strength a lot.
It looks a bit like there was some dirt within the fabric - however that's really stretching what you can (cannot) see on these images. If it were there for some time (not just from the last trip), that's bad too.
These might be totally false and your sling more or less fine, but again there is a fray and you have doubt, so retire it.