The thing with mathematics is that on one if you define something, you are completely free in choosing any name you want, and on the other hand you should find a meaningful name that evokes some intuitive ideas that coincide with what you actually wanted to do. So long story short, one thing that always bothered me is the following:
What is the origin of the term chain? When was it first used and what explanation (if any) was given?
To be a bit more precise, by chain I mean a finite linear combination of (oriented) simplices.
I looked into Dieudonné's "A History of Algebraic and Differential Topology" but there he only points out in a footnote that the term was coined by J.W. Alexander without giving any source.
Maybe related to this is the notion of a chain complex, which I think at least once someone explained to me as being a chain of arrows, eventhough these arrows are actually the things mapping chains to chains. This is of course not helped by the fact that some useful abstractions of a chain are called complexes (as in CW or simplicial). I guess that the term chain complex was coined some time afterwards, but in some cases concepts are found and given a name much later, so maybe there is a chance they are related.
Finally I may need to add that my background is in mathematical analysis, my primary interest in chains is integrating something along them. I am not very well versed in abstract algebra, so there may be some simple source that I missed. However I am more interested in the basic geometrical objects not in the abstract generalisations. (only there is no algebraic geometry tag...)