There are Google Chrome plugins that convert entire web pages into PDF, but are there any plugins or other software that convert only the selected area in a web page to PDF?
You did not specify an operating system. While this answer should work across Windows/OSX/Linux, specific program recommendations will be for Windows because that's what I primarily use.
To convert a selected area, well, there's two ways, both generally program independent. You can select the text and print the selection to the PDF printer, or you can take a screenshot, crop it and print the result to PDF.
Chrome has been known to cause users to struggle with printing only a selection. There's a few ways to do so, for example CtrlShiftP if you're on Windows. There's also a few plugins that allow printing only the selection.