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Could anyone recommend bibliography or name some useful methods to analyze which (exogenous) variables are most important in determining the value of a time series?

For context, I have a random time series $y_t$ (the sales of a product) along with several other series {$x^i_t:1\leq i\leq n$} that I think explain the sales. I want to know which of those "regressors" best explain the sales.

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That is a interesting question. At first I thought about R^2, but it seems not that easy. There doesn't seem to be a clear answer here. Here you can find some thinking approches, which consider different measures of importance:

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