It is very likely that the Silicon Labs Serial to USB driver has not installed properly.
I have always found this a pain in the butt and installation has to be done very carefully.
Did you plug in the Weather station in before installing WL? (Not a good idea)
Things to do: Go into Device manager and see if the Silicon Labs USB to Serial driver has installed properly. If it hasn't, you will see a yellow mark and an exclamation mark against it in Device Manager. This may be under Universal Serial Bus Controllers or "COM and LPT" If it's the former, you will probably see "Unknown Device" there.
See if under "COM and LPT" you have a "Silicon labs CP210xUSB to UART Bridge (COM3)" If you don't, or you have the yellow ! mark blob with it, then it won't be installed cleanly. (The number with COM may not be 3, but is in my case).
If you have the ! marks, you will need to completely uninstall it and reinstall it again
Go to Control Panel, bring up Programs And Features and look for Windows Driver Package - Silicon Laboratories.
Uninstall the package completely, uninstall Weatherlink, and then reboot.
Next, (making sure that the weather station is not connected to the PC before you reboot) re-install Weatherlink. This should cleanly install the Silicon Labs drivers too. Connect the weather stationa and you will probably be prompted/asked about intalling the drivers for the new device just connected (can't remember if that happens in W8 but I think it does..). Once that has completed, run Weatherlink and hopefully you should find your serial port.
Hope this helps... shout back if you need any more help.