Ian,
The two main parts to the Highcharts extension are the WeeWX skin that generates the json format data file and the html/javascript that displays the plot(s). WeeWX only real involvement is the generation of the json format data file, nothing fancy there and 3.9.1 should be compatible with the skin. The only issue I can think of here is perhaps the new skin config sitting hierarchy introduced in WeeWX 3.9.0, but that would likely only (at worst) result in unexpected number formats or units. If your json file is being generated without error it should be usable (a useful tool for checking your json output is one of the online json validators, I use
jsonlint.com, just open the validator page, copy the contents f your json format file and paste into the validator, hit the button and it gives you a yay or nay.
Given your description, it sounds like a javascript issue on your page. Easiest way to troubleshoot Highcharts is to through your browsers console/web console. where you can see any javascript errors. Where to find the console is browser dependent, I mainly use Firefox and it is under Web Developer -> Web Console or Web Developer -> Browser Console.Under Chrome it is under More Tools -> Developer Tools. Once you have the console open, clear it if it has any content and then force a reload of your html. Then its a case of looking for any (relevant) errors in the console output. Likely errors are a script (including Highcharts) loaded by the page being unable to be found, a script being unable to locate a file or Highcharts choking on incorrect data/settings in a file.
If your graph page is public facing if you are happy to give us the address I can look at the javascript console myself.
Gary