Joel,
The Drain is the only London Underground line to be completely underground. When train carriages are removed or replaced they have to be craned in and out through a hole.
The actual derivation of the nickname is lost to the mists of time.
I rather like the version put forward by one of my colleagues:
One theory is that this arose when the line was operated by train crew
in a link that otherwise operated normal surface suburban routes. In
comparison with working surface railways, the Waterloo & City consists
of underground tube tunnels. Messroom conversation would include
discussion of what turn a driver would be working tomorrow, and if it
was a Waterloo & City turn of duty, it was an obvious metaphor to say
that the driver was working "down the drain". Another theory is that
it was given this name by the maintenance staff, because the tunnels,
being under the river Thames, leak considerably allowing much water to
enter. This water has to be continually pumped out. This water gives
rise to a musty smell which provides a third theory for the name.
If you need translation help, ask.
After the name caught on with train crew it was only a matter of time before it was picked up by the traveling commuter public.