FYI, I'm going to be making my decision pretty early because I'm going out of town tonight (so basically within the next 4 hours).
So I kinda want to just pick Lonely Mountain because it made me laugh so hard, but I suppose I should consider some other options.
The other ones that struck me as possibly fitting the bill were Entwives, Weathertop, King of the Dead and Radagast.
I think Phantom and Nog are right in saying that Radagast really didn't want very much attention to begin with, so you can't really neglect something that wants to be alone.
The Entwives are a bit tough, cause when we hear about them it's more about the fact that they are lost and the Ents are trying to find them, but the reason they are gone could very well be because the Ents neglected them. But I think in the end the fact that the Ents seem to miss them so much means they are no longer neglected, so I guess they don't completely qualify.
Weathertop is a pretty good contender because of the state it was in. A truly valued watchtower would not be left to fall into ruin, so it was definitely neglected for a long time.
Then the King of the Dead (and all his men) were neglected for thousands of years, left in that weird restless state until Aragorn showed up and allowed them to be useful.
Soooo....I want to pick the Lonely Mountain for the humour factor (in the real life game humour is usually what I go with), but Weathortop and King of the Dead do have legitimate reasons to be chosen. I'll think about it for a bit.