The number one thing that I always recommend is PROACTIVE support rather than REACTIVE. By being proactive you can not only keep something in tip-top running shape, but you can also notify the client of the update or preventative task you performed.
While this is not only good for you and your servers, but clients will come to see this as a selling point and value for money. A small task such as clearing out huge log files can go a long way towards a happy client.
Of course, good communication is never a bad thing, and these days it's one of the biggest separators between all types of businesses.