GD as Corporation
Make the members of the GD purchase their votes via shares. This is a practical way to implement Montefeltro's observation that political power should be proportional to a member's "contribution", which is generally understood to be related to their economic output. It also allows some systems to effectively opt out of participation, and allows other more enthusiastic systems to gain more than they otherwise would.
Initially, all votes would need to be distributed to all systems in some person-agnostic manner. I would propose this be done on the basis of the blackbody temperature of the system. This measures the amount of energy available to the system, and indirectly, its mass. If the GD contains Kardashev Type I/II civilizations, then trying to measure anything less than this is futile and useless. If all the civs are < Type I, then I question how a GD is even possible, since they probably only have slow interstellar travel.
One the shares are allocated, they can then be traded on the open market. Of course, there should be more than one share per system during the IPO. A target should be something like 1,000 or 1,000,000 shares per system, average. This would allow members to adjust their share pool up or down by buying and selling a fraction of their representation (based partly on the level of their system exploitation...a a system with an energetic star but poorly habitable planets may decide that a more powerful civilization could better utilize its resources, and choose to sell many of its shares for a big economic bump, rather than trying to wield them all in galactic matters of only minor import to them).
There is, of course, the incentive for one or a small number of members to hoard votes. But who would they hoard them from? They have to buy every vote from someone else, and the more they amass, the more the rest of the GD will bid up votes, just like in any market. It would be irrational for any star system to charge less than the total economic output of the galaxy to purchase the last share, or even the share which achieves majority, whatever threshold that is. Every star system will be eyeing every other which is buying up shares, trying to determine their motives, their ability, their aggression, etc. If any system is able to corner a significant portion of the market, then it will only be because the majority voluntarily sold their shares. This means they either trust the majoritarian, or they are so economically/militarily/politically weak that they would have been steamrolled in any other system of gov't as well.
All of this discussion is moot without knowing what individual members are capable of w.r.t. power projection across the galaxy. Is there FTL travel? FTL signalling? How is trade conducted? What goods do members consider valuable enough to trade between stars? There must be trade, or there is no incentive for the most powerful system to simply conquer all of its weaker rivals. Unless, of course, the weaker systems are collectively powerful enough to resist the powerful one. But holding such a coalition together without trade relies 100% on game theory, and is likely very unstable. If even a few of the members decide that they are better off siding with the big dog, then the coalition falls apart.
What are the military capabilities of the member states? Can they destroy planets? Blockade them? How can they interfere with trade, given that space is so big? Can they kill all the citizens on a planet while leaving the infrastructure? Are the species remotely biologically compatible, or are they a hodgepodge mix? Are they at least all carbon-based, with some organic compounds in common?