From time to time when switching to primary weapon (1) I accidentally hit `, which in BF3 brings up the console. I don't see possibility to change that binding on the options screen.
Is there a way do change or disable that key binding?
It's a very very silly solution, but you can always pull the key off (unless you need it for programming or something):
Benefit is that you'll know if you missed the 1 key - disabling it can lead to confusion when pressing it does nothing.
My go-to solution for these types of problems is AutoHotKey, which can usually intercept almost every button press imaginable and remap them to other keys or disable them entirely. This doc in particular details how to write a quick script to remap keys.
For your uses, you might want a simple script like:
(that's the accent grave, two colons, and then the number one)
Which would change the accent grave key into the 1 key, thereby making it so that if you miss the 1, you still get the same effect. Install autohotkey, put that line in a blank notepad document, save it with the ".ahk" extension and double click.
(Note that if you have file extensions hidden in Windows Explorer, Notepad tends to save with .txt even when you don't want it to - if this happens, double clicking will open Notepad and not AutoHotKey. If this happens to you, put the filename in quotes in the save dialog, ie "battlefield-tilde-fix.ahk")
When you want your accent grave key back, just right click on the green H icon in your tray and choose Exit.
You can also get fancy, and make it so that it only effects the accent grave key when BF3 is running/has focus, but that's an exercise left to the reader :)
Some games have issues with AutoHotKey, so you might also need to do something more advanced, this is covered in their FAQ.