How can I optimize a mysqldump of a large database?
The main bottleneck in the dump like this is drive I/O. You are reading a load of data and writing it again. You can speed this up in a number of ways:


Make sure your output is going to a different drive(s) than the one(s) the database files are stored on - this will make a massive difference with spinning disks as the drive heads will not be constantly flicking between the location being read from and the location being written to.
The output of mysqldump will be very compressible, so if you can not separate the output from the input as mentioned above pipe the output through gzip or similar. This will reduce the amount of writing being done (so reduce the overall IO load, and the amount of head movement) at the expense of some CPU time (which you may have a lot of spare at these times anyway).
Also, (as well or instead of compression) pass the output through a pipe utility (like pv) that supports large write buffers to group blocks written to the drives together more, again to reduce the effect of head-movement latency - this will make quite a difference if using the --quick option to reduce the RAM impact of backing up large tables).
Only run your backup process when IO load is otherwise low.


You may be fixing the wrong issue though: it might be easier to address the connection drops instead (though reducing the I/O load imposed by your backups will help reduce the effect you have on other users so is worth trying anyway). Could you run your manual backups through screen (or similar tools like tmux)? That way if your connection to the server drops you can just reconnect and reattach to the screen session without any processes getting interrupted.

If you are sending the data directly over the connection (i.e. you are running mysqldump on your local machine against a remote database, so the dump appears locally) you might be better off running the dump on the server first, compressing as needed, then transferring the data over the network using a tool (such as rsync) which supports partial transfers so you can resume the transfer (instead of restarting) if a connection drop interrupts it.

As part of your "reducing the size of the overall database to resolve this issue" I would guess that a large chunk of your data does not change. You might be able to move a large chunk of the 1.2Gb from that main table off into another and remove that from those that are copied by the mysqldump call. You don't need to backup this data every time if it never changes. Splitting data between tables and databases this way is usually referred to as data partitioning and can also allow you to spread the data and I/O load over multiple drives. High-end database have built in support for automatic partitioning, though in mysql you will probably have to do it manually and alter your data access layer to account for it.

Straying off-topic for this site (so you should probably nip over to ServerFault or SuperUser to ask if you need more detail): If you seem to be losing connections due to inactivity, check the options in your SSH server and SSH client to make sure keep-alive packets are enabled and being sent often enough. If seeing drops even if the connection is active you could also try using OpenVPN or similar to wrap the connection - it should handle a short drop, even a complete drop if your entire connection is down for a few seconds, such that the SSH client and server don't notice.