First of all, welcome to the forum!
I've stayed at a number of sites over the years; here's a selection:
Wasdale Head - the field opposite the Inn, not the National Trust site amongst the trees. Not the quietest of spots but amongst some of the most dramatic mountains in the Lakes. It is still pretty cheap.
Turner Hall Farm, Dunnerdale (a mile north of Seathwaite) - there's a special atmosphere about Dunnerdale, and good walking from the site. Also - Newfield Inn is just a mile down the road.
Moss Howe Farm, Witherslack, in the South East Lakes - stayed here for a week in July, a really chilled out site in a lovely spot. Camp fires allowed! Lower level walking in limestone country. Food and beer in nearby Derby Arms excellent.