A portable way to do this is to maintain a map of names, keyed by the thread’s ID, obtained from thread::get_id(). Alternatively, as suggested in the comments, you could use a thread_local variable, if you only need to access the name from within the thread.
If you didn’t need portability, then you could get the underlying pthread_t from thread::native_handle() and do whatever platform-specific shenanigans you like with that. Be aware that the _np on the thread naming functions means “not posix”, so they aren’t guaranteed to be available on all pthreads implementations.