I recently bought LG G3 and I have enabled debugging mode but Eclipse is still not recognizing it, back when I used the Galaxy S3 for development I had to install Samsung Ki
I also had a great deal of trouble when attempting to debug android applications on my shiny new LG G3. What ended up working for me was first installing the default google drivers onto the G3 and then following the process detailed by @lilbyrdie.
The process of installing the google usb drivers is very nicely gone through here.
However, with the most recently pushed android update to the G3 there are no longer the options for internet connection or charge only; but they seem to have fixed this defect forcing you to make this odd choice to enable debugging. The only options remaining are MTP and PTP, but if you pull down the notification drawer, choose the 'USB connected' notification (assuming the device is connected to your computer via USB of course) and from there click on 'Media sync (MTP)' even if it is already selected then debugging should be enabled.