Using lambda calculus to derive time complexity? Are there any benefits to calculating the time complexity of an algorithm using lambda calculus?  Or is there another system designed for this purpose?

Any references would be appreciated.
There are quantitative results concerning $\lambda $-calculus, in the form of measuring the length of reductions in (typed-)lambda calculus. But this is of course far from saying anything about the complexity of algorithms (especially that the bounds obtained are fast-growing). See for example:  Arnold Beckmann, Exact bounds for lengths of reductions in typed $\lambda $-calculus, Journal of Symbolic Logic 2001, 66(3): 1277-1285.

For something closer to your question, there is a current project that develops and studies a type-system (a functional programing language) which by static-analysis can determine (polynomial) run-time bounds of programs (as well as other resources used by programs). So in some sense, this might hint that there might be some advantage in using functional-programing for analyzing  run-time complexity. The project homepage is here.

A possibly representative paper of this project is: 
Jan Hoffmann, Martin Hofmann. Amortized Resource Analysis with Polynomial Potential - A Static Inference of Polynomial Bounds for Functional Programs.  In Proceedings of the 19th European Symposium on Programming (ESOP'10).link