Does the yamulke at a Jewish funeral have to be black?

No. By default, there's a traditon of somber clothing for funerals, and this suggests that a black, dark grey or brown or dark blue kippah (yarmulke) at a funeral should be the default. Usually, funeral homes stock black, and usually, synagogues stock black, but if the deceased left behind a directive asking for a different color and provided a stock of kippot in that color, I doubt that the request would not be honored. Note, however, that Jewish tradition asks that the funeral be as soon after death as is feasible. This means that elaborate funeral preparations are generally out.