You can also auto-complete based on the command history. Start typing, then invoke the keys which are bound to ed-search-prev-history and ed-search-next-history. This applies if mysql comes with libedit support. The default keybindings are Ctrl-P and Ctrl-N, but this can be customized in .editrc. My example for Ctrl-up and Ctrl-down:
# start typing, then press Ctrl-Up
bind "\e[1;5A" ed-search-prev-history
# start typing, then press Ctrl-Up, then Ctrl-Down
bind "\e[1;5B" ed-search-next-history
Previously, mysql was based on readline, and then history-search-backward and history-search-forward are the correct commands. Configuration then was by means of .inputrc. Same example as above:
# these are the key bindings for the readline library
# start typing, then press Ctrl-Up
"\e[1;5A": history-search-backward
# start typing, then press Ctrl-Up, then Ctrl-Down
"\e[1;5B": history-search-forward
So, say you started typing sel and invoke Ctrl-Up, select * from some_long_table_name would come up if that is a command I have used earlier.
On OS X 10.11.6 I set --auto-rehash as described above, but it did not work. (This is OS X so mysql is compiled with the BSD libedit library.)
Then I remembered that I had set vi key-bindings for mysql client by creating ~/.editrc, containing one line: bind -v. This works great for giving me vi-like navigation in mysql client, but it broke column name completion (I was able to verify this by removing .editrc).
So I researched a little bit and found that ~/.editrc should have at least the following lines:
bind -v
bind \\t rl_complete
With this additional line, name completion works correctly in mysql AND vi-like navigation works also. (There are other .editrc settings which greatly improve mysql client navigation, but this isn't the place to start that thread of discussion.)
I know this is an old question, but I've found very helpful MySql cli client with advanced autocompletion:
mycli. It's much smarter than builtin auto-rehash feature.