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In a linear regression model, I'm considering adding in two control variables: One for population density and another measure that assesses vehicle ownership. These variables are correlated with each another, which would cause problems with multicollinearity. Neither variable is even close to being statistically significant in the model when only one is added, but both become statistically significant if they're both added to the model.
Any recommendations for what I should do?
Based on the information that you have provided, you should include both. My guess is that one of the variables is suppressor variable.
Also, I really doubt you have any multicollinearity, as multicollinearity tends to manifest itself by existing variables ceasing to be significant when a new variable is added. (Note that two variables being highly correlated is not multicollinearity; check out one of John Fox's books on regression if you want to know more about this.)
If you are adding the variables as controls, it is completely irrelevant whether you have multicollinearity or not (unless the correlation between the variables is 1). It is only a problem if the multicollinearity is stuffing up one of the effects you are interested in estimating.
Check the variance inflation factor (VIF) scores to see the strength of multicorrelation. The VIF would indicate by what degree the standard errors are inflated; meaning you would probably NOT have gotten a significant result if there is strong multicorrelation.
If you use those two variables as controls, check the VIF on the variable of interest. If there is no strong multicollinearity, then there should be no problem.
