Hi @Anonymous and Welcome to the O2 Forum.
Sometimes turning the phone off -> Remove the Battery -> Re insert Battery {Handset Model Dependant on having a removable battery}, and turn the phone back on is known to remedy lots of phone glitches.
With all phones a procedure can be carried out called a Hard Reset. This can rectify glitches and especially after any software updates. As this procedure typically deletes your saved data in the phone memory its advised to move all that you can to any available SD Card (if applicable) ,and perform a back up.
This is one of the last things to try.
Link to how to perform this is :
http://www.hard-reset.com/samsung-galaxy-s3-mini-hard-reset.htmlI've read in a forum here
http://forums.androidcentral.com/samsung-galaxy-s3/301017-s3-home-button-not-working.html that this may help as an interim workaround :
https://play.google.com/store/apps/details?id=com.smart.swkey.nonrootIf a Hard Reset or any other tips you find by a Google search don't work , then it could be a hardware issue.
In this case I'd suggest contacting the Manufacturer by their Uk Home page here
http://www.samsung.com/uk/support/ to enquire if they can help.
O2 do have Repair Services and you just drop the phone off at your local O2 store ,and they'll send it off to their own repair centres.
As the Manufacturer knows their Hardware / Software better than others its recommend you use them as a first port of call.
Let us know how you get on.
