I know the bulk of the material is in the anode and cathode and that the electrolyte only makes up a small fraction of the total weight. I'm interested in a quantification of this.
You need to know the chemical composition of the anode, cathode and electrolyte. Then you need to know the energy density and charge capacity of the battery. Considering these data and the Faraday Laws it is easy to answer to your question. See K Ozawa,ed, Lithium Ion Rechargeable Batteries, Weinheim, Wiley-VCH, 2012.
If you would like to have a rough estimation the above mentioned sources should be fine. Though, if you are planning precise calculations there's many things you have to take into account. Just a quick overview, where those numbers come from:
Since the electrolyte is just an expensive tool which ensures the battery functionality, it's recommended to reduce the amount as much as possible for financial reasons. Again, using too little electrolyte can increase internal resistances due to low conductivity, increase overpotentials due to Li ion concentration gradients or even lead to a shutdown (capacity loss) if the solvent is consumed by side reactions and therefore the cell dries out. If you are not interested in cost optimization, you'd better use spare amounts of electrolyte.
Balancing the electrodes again is a complicated topic, since there are quite some challenges. Electrode properties depend on active material BET-surface areas & kinetic properties as well as electrode porosities and tortuosities etc. People generally try to equilibrate the energy capacities of anodes and cathodes, so that no active material is wasted. Though, electrode active surface area and kinetics lead to an undesired overvoltage distribution, which can cause critical side reactions.
For instance, underdimensioning the anode would cause a small anode surface area. To achieve high currents during the anode lithiation, the anodic potential reaches critical values due to a high overpotential demand. This low potential can cause lithium plating & dendrite formation, which again is a huge safety issue.
Summing up: Electrolyte balancing is an easy task. Electrode balancing requires lots of material and half cell investigations.. and a bunch of experienced people.
I do agree with these general comments, but the previous ones were of more practical help. Of course nowadays there are already some commercial lithium batteries and clearly the nature and composition of electrodes and electrolyte would change the correct fractions used.