There are a number of foods or food 'types' that can be used to colour e.g. tumeric, beet juice, saffron, paprika and so on BUT the trouble with cotton is that none of these will be washfast as such. I read a great piece a year or two back that debunked the claim some people make of garments being 'naturally food dyed' and so on. It's rather like the colourings that are used to make glace cherries red because without the colourant they would be beige and so on. Many of those 'naturally food dyed' claims are really about the typical food colourants used in industry. That said, it's always worth contacting say a hemp sales place (hemp yarns etc) to ask how they 'fix' their dyes when they claim to use 'natural' dyes. See what they say.
Most very natural dyes will fade and streak (be careful washing with other items) BUT this link claims the process to be colourfast. It combines everyday common dirt basically (but remember, many indigenous peoples use ochres for painting) with soymilk and it's the soy that is claimed to be the 'binder' between the dye of the ground dirt or stone and the cotton material. Soy is put through a manufacturing process so it may not be quite as 'natural' as we think BUT this appears to be worth a try for you. Let us know how you go with it if you try it out??
As another tip my mother and grandmother used to always add salt to a new garment as it was first washed (add salt to the water) claiming it would 'hold' the colour.