So, as somebody who worked on the Hill before email, I can tell you what happens at the back-end. Ninety nine times out of 100, your Congressman doesn't see anything you write, hand-written, emailed, phone called, or whatever. The person who you are influencing isn't really the Member, but rather his LA (Legislative Aide or Legislative Assistant see here how a Congressional Office works). Offices are pretty much always set up the same way, and the grunt work of reading the mail and tailoring the boilerplate response is the work of the grunts.
What the grunts do is to gauge opinion so that the Member knows how his district feels about a thing. It isn't as scientific as polling, but that's not the point, because it measures intensity. Likewise, if the Member is getting 368 pieces of correspondence for a thing and 369 against, that isn't necessarily going to sway the Member. Instead, the point is this - When the Member asks the LA, "What's the district like?" The LA is going to subjectively characterize the response.
Now, on big issues, the answer will be "A lot of mail, and its running 50-50." But on most issues, its going to be, "Well, not really a lot, but I seem to remember..." And it's that last bit where your method of contact really matters. If I got a phone call, sad to say the pace of the place is such that it was in one ear and out the other. I got a typed letter, the first time it made an impact, but further copies of the same letter said "Oh, these people xeroxed it." (That was when we still had xerox'ing and not googling!) I imagine email works very, very much the same. But, when I got a handwritten note, I almost always said, "You know, this person really cares." I can't say that was worth 3 or 4 typed things, but it made an impression. And when the member asked, I would say, "I remember ..."
Oh, and if it was written in kindergarten letters or had drawings or otherwise was clearly a kid, I might have had occasionally said to the Member, you need to see this one. And yes, it really does have an impact then.
Studies will, I'm pretty sure, not be done on this. From an outcome point of view, Members wouldn't want to think that the method of communication matters. And, really it shouldn't. But we are human beings trying to deal with a tidal wave of communication. We are finite. And its only the extraordinary that comes out on top.
A handwritten note carries more impact because of its rareness. That's really the only the only thing that matters.