Welcome to the MDK forum! And please accept my sincerest sympathy for you and your friend. She is blessed to have you in her life.
Yes, bamboo or silk will be a much softer yarn than just about anything else. There are lots of chemo hat patterns available for free on Ravelry.com Don’t be surprised if they suggest a smaller size than you’d normally make for any adult, since your friend will be wearing it on a bald scalp. I suggest a very simple hat, since she will likely sleep in it, too.
And gosh, if you go onto any yarn selling site, or to a nicer yarn store (not a big chain craft store), you will definitely find bamboo or bamboo mix yarns in a variety of colors. Take a look at Knit Picks - their Luminance is 100% silk, and their Galileo is a 50/50 wool and bamboo that is very soft.