Can't Type Tilde character in Mac OS X

On a keyboard with that "<>|" button between the left shift and Z, holding down Shift and pressing that button works on my end. Neither Alt-5 nor Alt-N do the trick on my end.

If someone else is still interested in how to swap the § key with ~, I use Karabiner to do that and it's pretty easy to use. Just look for that option which is included in the default ones among others.

It can also be used to remap other keys if you want. For example, you can use hjkl keys to emulate the arrow keys (vim users will love it).

Don't know if you are still reading this but if you download a program called Ukelele, you can make custom keyboard maps.

Then you need to put the keyboard map into ~/Library/Keyboard Layouts/, log off and then back in.

Once you have done this you can choose your custom keyboard layout and you will be able to use the ~ key without any problem (I'm using a Logitech illuminated keyboard using the default Logitech-UK keymap that comes with Ukelele and everything works).