This can be solved using synclient:
Type synclient -l to list your configuration. At the end of the output, you will find something like
RightButtonAreaLeft = 3068
RightButtonAreaRight = 0
RightButtonAreaTop = 4326
RightButtonAreaBottom = 0
These values have to be changed to 0 (zero) by typing these two commands:
synclient RightButtonAreaLeft=0
and
synclient RightButtonAreaTop=0
Clicking anywhere on your trackpad will now always trigger a left-click. To right-click, do a two-finger click on your trackpad.
To make these changes permanent, write the two commands to a shell script and add the script to your startup applications:
nano ~/.synaptics-custom-settings.sh
Paste this code to the shell script:
#!/bin/bash
synclient RightButtonAreaLeft=0
synclient RightButtonAreaTop=0
Open the Startup Applications program in Unity and add ~/.synaptics-custom-settings.sh to the list of startup applications.
I've got an xps 13 and have had issues with the trackpad also. I managed to fix it by complete accident the other day, you can try this and see if it does the job for you too: go to your 'additional drivers' application from the launcher, look through and see if you have an unknown driver, enable it. The driver will say it's from the 'sommerville's dino platform'. I have no idea what that is exactly, but it's made the trackpad work perfectly!
Best Answer
This did it for me: http://kernpanik.com/geekstuff/2015/01/12/disable-rightclick-synaptics.html
For me this disabled the right hand side click area without disabling two finger "right-click".
Essentially, run this:
To persist, add a script that does this into "Startup Applications".