I\'m currently working with a fairly old product that\'s been saddled with a lot of technical debt from poor programmers and poor development practices in the past. We are star
Managers care about making $ through growth (first and foremost e.g. new features which attract new customers) and (second) through optimizing the process lifecycle.
Looking at your problem, your proposal falls in the second category: this will undoubtedly fall behind goal #1 (and thus get prioritized down even if this could save money... because saving money implies spending money (most of time at least ;-)).
Now, putting a $ figure on the "bad technical debt" could be turned around into a more positive spin (assuming that the following applies in your case): " if we invest in reworking component X, we could introduce feature Y faster and thus get Z more customers ".
In other words, evaluate the cost of technical debt against cost of lost business opportunities.