I am going to approach this question differently. Since the question is marked with the rock-climbing tag the use of a 7.5mm Technora escape rope that AM_Hawk's answer focuses on seems unlikely. Also, the method by which he arrives at a minimum Prusik cord size doesn't address certain issues.
All else being equal a smaller cord for a friction knot, such as a Prusik, will grip better than a large one. I have seen several guidelines, such as use cord no larger than 70% of the diameter of the rope being ascended, but truly this is only a guideline. A stiff and slick cord will have much less purchase than a limp and rough one, therefore maximum diameter is determined pragmatically by what will grip. There are cords made specifically for friction hitches, and these typically limp cords will work better even in larger diameters than will stiff accessory cord not intended for the application. For example, BlueWater 8mm Dynamic Prusik Cord is limper and will grip better than their 6mm Accessory Cord.
As far as minimum size, there really is none from a friction standpoint as the smaller cord grips better; it is entirely possible to use a shoelace, paracord, a string from your chalk bag, etc., so long as it can hold your weight. One would be a fool to have these thin cords be his only attachment to the rope of course. You can protect the ascent by progressively clipping in to loop knots in the rope, as AM_Hawk described, or you could use a non-toothed progress-capture device such as a Petzl Shunt. (Please don't implement a system without a full understanding of the devices, failure modes, and risks involved; this is merely an example.)
Perhaps you are seeking a minimum size of friction hitch that is safe as a primary connection point. That is hard to answer. One possible failure mode is burn-through; that is, if the knot slips the friction will melt through the cord long before you deck. Hypothetically a smaller cord, if the loop is appropriately sized and the knot well dressed, might be less susceptible to this as it should be less likely to slip, but a larger cord stands a better chance of arresting the fall before it melts or tears through. There is also the consideration of fusing. Several test reports describe the friction hitch fusing to the main line after slipping; while this is less immediately terminal than burn-through it could still leave you stranded.
I was unable to find dynamic (drop) testing of small diameter Prusik cords. However, here are some resources that seem relevant.
A test report for twin 8mm Prusiks at rescue loads: CMRU July 1997 Drop Tests.
An extensive test report for using a Purcell Prusik as a personal anchor, including 6mm cord: Rigging for Rescue -- Lanyards Part II: An Examination of Purcell Prusiks as Personal Restraint LanyardsRigging for Rescue -- Lanyards Part II: An Examination of Purcell Prusiks as Personal Restraint Lanyards. (The circumstance is almost entirely different from rope ascension: the fall factors are much higher but slip is inherently limited.)
Drop testing on arborist's equipment: Ropes and Friction Hitches used in Tree Climbing Operations -- Paolo Bavaresco, Treevolution