I believe that dying languages are worth saving because it helps with keeping a unique aspect of a culture thrive. Also, many dying languages used to flourish in the past but are slipping out of use today? Take Latin for an example; no country in this day speaks Latin anymore, but historians in Europe should know the basics of this language to read and understand old scripts in Roman and Medieval times. Even though a language may be dying, it's still worth it to try and save them.
Many of the smaller, endangered languages have these really cool features that set them apart from all other languages. These features are really interesting because they demonstrate just how powerful the human brain can be at exchanging and interpreting spoken information. Ultimately, the choice of whether or not they should be saved rests in the hands of those that speak it, but I do believe more should be done to preserve them in an archive of sorts where future linguists can examine them in the future. Unfortunately, there are some language groups out there that are extremely overly protective of their languages and would rather them die out than have others try to learn them and bastardize the language.. There are a few Native American tribes that are like this.