You can use the particle instance modifier to stretch a mesh object onto a particle hair strand, as discussed here: Hook/parent/constrain to particle hair?
However, every strand within the same system gets the same mesh. I am looking for a way to vary the size of the strands without making multiple particle systems. Is there some way to non-destructively modify or distort the object so that every strand isn't so uniform?
UPDATE: Starting bounty. As mentioned in old comment, i'm looking to vary the thickness. The length can be varied within the modifier settings.