I've wondered this myself, because the idea that all they did was hole up and hide didn't sit well with me. If they weren't willing to give their lives for this cause, they wouldn't have joined the Order in the first place, so I doubt that they stopped fighting even when the danger increased.

So we know they were doing something.

Do you think Kingsley was the leader throughout DH?

I'm too lazy to go look at DH to see who was the leader. But if memory serves me right, it seems that you could probably have free-reign with this. I can see him being the one that they might look too after Dumbledore and then Mad-Eye died.

Do you think they were attempting to infiltrate the Ministry and try and take over control again?

For some reason, I don't really see this, maybe because there aren't a lot of known members of the Order already in the Ministry (besides a few Aurors and Arthur, of course, but none of them would have had enough power). Also, the Ministry wasn't much help to the Order before it fell, I'm not sure they would waste their efforts trying to get it back (even if they could, which I doubt). And I think that too many of the Order that were in danger with the Ministry which would limit their abilities because people would recognize them (Percy, anyone?). Another reason is they were dealing with a lot of the aftermath (the Muggleborns in the like), and would probably have their handfuls with that. Also, I think they would know to get the Ministry back they would really have to take out Voldie first. So, I suppose there is a lot of reasons that I don't think that they would try this.

Would they help Muggle-borns flee the country?

I really like the idea that they would have some sort of system to get Muggleborns out of the country. Many of them probably sheltered Muggleborns as well. (an Underground Railroad, sort of thing, perhaps). I can definitely see Fred and George hiding some in the basement of their shop. I can honestly see this the center-point of what they were doing. Such a thing would equal out to a LOT of secrecy, which to someone outside of the Order, would actually look like the inactivity we see.

Of course, you would have to explain why they didn't help Ted Tonks get out of the country if you wanted to use this angle, but of course, there is always the explanation that Ted had to go on the run very quickly (DE showing up at the house - Bellatrix certainly knew about him) and the Order didn't have time to reach him to get out. As long as you explain that, you can make a good case that that was what the Order was up too.

They probably would have also done whatever they could to 'help Harry'. Speaking they didn't know what he was doing, this could have been anything from 'attacking Death Eaters' to weaken the Dark Lord's forces or attempting to spy at the Dark Lord's plans (since they lost Snape, their only spy as far as we know - though there could be good argument that there were more spies already in place). Maybe they even captured a random DE to interrogate him/her.

Since there is so little information, canonically you can do just about anything. Just because Harry didn't hear of it, doesn't mean it doesn't happen. His sole source of info is from the Pottercast and I don't think they'd be ratting themselves out.