If it's a Nikon strap emblazoned with branding please abandon it and get a plain one complete with all the required bits ...
Cover the logos with black gaffer tape and/or attach the strap upside down. Gaffer tape can also tidy up the loose strap ends and hold loose rubber grips in place when they start peeling off camera bodies.I quite fancy the idea of a more subtle one. Looks a bit silly with D600 all over it. Any recommendations?