As a fan of simple approaches I would start as simple as possible. Definitely any method based on time-boxing isn't an option as they expect steady input flow of work. From your description it isn't the case.
It seems that Kanban would be a very good match for your situation.
First, it bases on pulling work, which basically means that the team adopts the way they work to reacting to changing scope of work. In short, they start working on whatever is the most important thing at the moment, but at the same time they limit work in progress so they can easily react to rapidly changing priorities.
Second, it gives you visualization, which not only is a good tool to assess current status of work but also helps you to change and improve the process you follow. Visualizing work, possibly on a Kanban board, would give you quick high-level overview of a project pretty much any time.
Third, there is measuring the flow. With simple measures like cycle time or lead time you can learn how fact you react to new tasks which appear in the queue and how much time you need to get them done.
On the top of that, Kanban is very easy to implement and adds very little hassle to use it, especially that with Kanban you don't change your initial process. You work as you used to, and then improve gradually as you learn more about the way you work and the work itself.
If you want to read more here is a description of Kanban principles.
And there is free minibook which can be downloaded presenting Kanban more extensively (and using Scrum as background).