A friend of mine wanted to scent her sheets with lavender oil. That makes a lot of sense. Lavender promotes relaxation, so it is a great scent for bedtime. Plus, it is all natural, so there are no worries about sleeping in chemicals or synthetic fragrance.
We thought of putting oil in the rinse, but it would take a lot of oil to leave a noticeable scent on that amount of water and fabric.
Another possibility would be to put a drop or two on a cotton ball and put it in the dryer...but then you risk oil spots on the sheets, and the heat would volatilize (evaporate) the scent very quickly, especially for a delicate top note scent like lavender.
My favorite solution is the old fashioned "pillow spray". In a 2 oz fingertip pump spray bottle put 1 oz of filtered water (not chlorinated tap water) and 1 oz of vodka (as a dispersant and preservative) and 10 - 20 drops of your favorite essential oils. You can use one oil, or a blend of two or three oils. Obviously if you are going to use the spray on your pillow, you want to stick to relaxing or meditative scents like lavender, sandalwood, or ylang ylang. Bergamot is a good all-around stress buster, and can help a sleep blend if worry is keeping you awake.
For pillows and your sleep area, avoid very activating or invigorating scents like citrus and strong floral scents like geranium and rose. While they do really nice things for your mood, they may distract from sleep.
If you don't want to use vodka, just use 2 oz of water, but be sure to shake the bottle very, very well before using and frequently between spritzes. Without the alcohol preservative, be sure to store the bottle in a cool dark place. The shelf life might be reduced without the alcohol, but hopefully you will enjoy and use the mist enough that shelf life won't be an issue.
Use with caution on delicate or water sensitive fabrics, like satin pillowcases and such.