Ours was also determined by popular vote of the residents with the PD able to veto if necessary. I believe the veto was primarily to prevent someone who was felt to be too far behind academically from taking on additional responsibilities instead of studying/working more. Two twists to our system, both of which I thought were good ideas:
We usually had 2 co-chief residents. If done poorly, having 2 could be pointless, but we tried to do a reasonable job of dividing duties between them in a fairly logical/organized fashion so the overall responsibility was a bit less onerous and only one person would consistently be the "go-to" for any given issue/topic.
Chief resident(s) served during 3rd year. This was great because it did not interfere with board study time of 4th years and it was early enough that you could put it on your CV when applying to fellowships, which most people do during 3rd year. I'm not sure how much fellowship directors care about it, but, I would think it could theoretically be a bit more significant on the CV at that earlier point in one's career vs. when applying for actual jobs.
I'll second the opinion of many others that it is probably more of a pain in the butt than it is a valuable/useful experience for most, but YMMV. If I remember correctly, the only meetings our chiefs had to attend that the rest of the residents were not invited to were resident selection committee (for discussion/ranking of applicants), where they were expected primarily just to share the opinions of the residents as a whole. Serving at/Attending meetings of various hospital/professional society/resident committees was not limited to or necessarily associated with being chief resident in our program.