Why was '~' chosen to represent the home directory?

Quoting Wikipedia:

On Unix-like operating systems (including BSD, GNU/Linux and Mac OS X), tilde often indicates the current user's home directory: for example, if the current user's home directory is /home/bloggsj, then cd, cd ~, cd /home/bloggsj or cd $HOME are equivalent. This practice derives from the Lear-Siegler ADM-3A terminal in common use during the 1970s, which happened to have the tilde symbol and the word "Home" (for moving the cursor to the upper left) on the same key.

You can find photos of the Lear-Siegler ADM-3A keyboard on this site.

This terminal is also the source of the movement commands used in the vi editor: h, j, k, l for left, down, up, right.

The Home key was also used for the tilde character on old terminals. See here for more details.