I've never used a monitor for my documentaries, but I'm now shooting a fiction piece and the director wants to use one. I think its just to see whats going on during shooting.

How are monitors powered on an outside shoot? And whats the best cable to feed the pictures (for a Sony Z1)

(I guess if he just wants to follow the action an old tv or laptop could be used?)

To focus HD, you want at least an 800x480 resolution, component inputs, and you'll need a battery of some sort to power it in the field. The options out there tend to be a bit pricey or lacking in features in my opinion, however, there are a lot of brands out there to choose from. It's discussed a lot in here.