Say we have a planet that takes 8 Earth days to fully rotate on the axis. At a latitude of 45 degrees, during the winter solstice, its day lasts 1 Earth day, or 24 hours, and its night lasts 7 Earth days, or 1 week. What axial tilt will result in a day length of 1/8 of the axial rotation at an latitude of 45 degrees?
Looking at Earth, even when you get to a latitude of 60 degrees there are still about five and a half hours of daylight (ie almost a quarter of the day length). In order to get the daylight length down to 1/8 of the rotation period, it is necessary to be within a degree or so of the Arctic/Antarctic Circle. Which puts the Arctic/Antarctic circles at 46-47 degrees north and south respectively, with a corresponding tilt of 43-44 degrees.
Exactly how close depends on the definition of "daylight" - does the sun need to be above the horizon for the entire 1/8 of the rotation or does a period of pre-dawn/dusk light count? Is there a mountain range immediately to the north/south as applicable that will block low-angle sunlight?
Note that this puts the tropics within 2-4 degrees of the arctic/antarctic circle! It may be worth looking at the answers to this question about a planet with an axial tilt of 60 degrees to consider the impacts on your proposed planet's environment, even though it is not quite as extreme.