Ross Kirsling's D3.js-based Directed Graph Editor is a simple and convenient point-and-click online graph editor. It doesn't include weighted edges, but it probably wouldn't be difficult to add that capability if you're willing to learn D3.js. (Learning D3.js's weird conceptual model is not necessarily a trivial project, even for experienced programmers). Kirsling's graph editor is not very smart about layout (although that might be fixable), is not designed to load data (although that's easy to add in D3.js), and it lacks other fancy features that some of the tools listed here include. However, it's right there online, and it might be one all needs for some purposes. Kirsling's graph editor is part of his Modal Logic Playground.
Ben Guo's Force Directed Graph Editor, also in D3.js, is a similar tool. The graphs aren't directed, but this tool is better at layout.
EDIT:
(1) Searching on various combinations of "D3" with "directed graph" and "edit" brings up a few more options.
(2) On an iPad interation with D3.js programs sometimes doesn't work well--at least in Safari. The programs mentioned above work fine on a computer, but I couldn't interact fully with them in iOS.
However, for iOS users, Instaviz is a great little app. Although there are many features it doesn't have that I wish it had (I think it's still being developed), what it does do it does very well and pretty easily.