The Family Ness was written and created in 1983 by Peter Maddocks, who also created Penny Crayon and Jimbo and the Jetset - just two other classic 80s cartoons.
Not just one, but a whole family of Nessies live in the depths of Loch Ness, Scotland. The monsters all know that they must have nothing to do with humans, until the day they meet Elspeth and Angus. They are the two children of Mr McTout - the keeper of Loch Ness (and a non-believer).
The main thing I remember, though, was the theme tune and how awesome it would be if I knew Family Ness!