Working with just one light is a great way to create moody portraits that can be full of character. It's a perfect technique for shooting subjects who are a little older as low light can really exaggerate lines and wrinkles but don't let this put you off photographing low light portraits of younger members of your family. Shots of kids converted to black and white or shots of women in candle light can be really atmospheric. Just remember to have your tripod to hand as you'll be using long exposure you won't be able to hand-held without it looking like you took your shot in the middle of an earthquake.
When it comes to picking a light source a studio flash is always an option but if you're working from home try using a torch, light from a window or a table lamp to add a little light to your scene. If you find the light's a little too harsh, try moving your subject further away from it or if you're using a window, diffuse the light with material such as muslin or parchment paper. If you're using flash try fitting a softbox or use barn doors to direct the light to where you want it to be.
Having the light to one side of your subject will mean one side of their face will be really bright while the other's hidden in shadow. For something less dramatic use a reflector to bounce light into your shot, adding detail where it was originally lost. If you want to add more light move the reflector closer to your subject and experiment with different reflector shades to change the colour balance of the light. To create really strong shadows try positioning your light source under your subject. Just be warned that this won't work with everyone!
One final note: Don't take your exposure reading from the dark part of your set-up as this will cause the lighter parts of the image to appear overexposed.