Look at the meanings that all the relevant people understood the words they were using to have. That's why, for example, judges will go to the legislative history of a statute, in order to understand what the legislature thought they were doing when they passed an act.
That's also how judges evaluate whether there was a true "meeting of the minds" in order to determine whether a contract is valid; if there is no "meeting of the minds," then the contract is not valid, regardless of the words written on the paper.
Fortunately, that's not the sort of advice that I would need to give, since it's already more or less common practice. You see legislative histories checked a lot more often than dictionaries.
It's very rare you need to resolve a debate "over the meaning of a word." In my experience, the person who needs to determine the exact meaning of a word (as opposed to the common understanding that everyone has) is usually a person clearly in the wrong and trying to escape through a loophole. Usually the semantic debate is a smokescreen to hide the real issue.
Case in point : the near-infinite number of idiots who come onto this forum and declare the Randi Challenge is unwinnable, since by demonstrating the "paranormal," it suddenly becomes "normal" (and therefore ineligible for the prize). That line of argument unfortunately ignores both the fact that Randi has stipulated (in advance) a laundry list of things that are paranormal by definition, and also that the proposal drawn up simply specifies what the claimant will do. Even if Randi were to try such a loophole, any judge would have no problem recognizing that as a pathetic smokescreen, because the substance of the dispute would not be what "paranormal" means, but whether or not there was a "meeting of the minds" and whether the claimant had performed as agreed.
