I do not know who first came up with the term "handwavium," or even if it originally had to do with science fiction. I first learned it at SFConsim-l, the best site for space discussion that I know of.* Handwavium, and its cousins such as unobtainium, are all the materials, technologies, and new laws of physics that allow us to do fun things we cannot do now (such as travel in space for less than a fortune).
* The name stands for SF Conflict Simulation, i.e. wargames, so it is probably not a place for the confirmed pacifist, but by no means is discussion confined to futuristic ways to blow each other up.The queen of handwavium is of course FTL (Faster Than Light, for those of you who are not SF geeks) - a shameless violation of relativity, almost universally used in science fiction so that we can travel among the stars without decades or centuries of boredom en route, and get home in time to bore our friends while they are still alive. Lesser forms of handwavium are required for such conveniences as "torch" drives, rockets that can run at full power for weeks at a time, and a host of other gadgets that make up the standard space SF kit.
As space SF subgenres go, rocketpunk gets by with less handwavium than most, or at any rate with subtler handwavium. Rockets, after all, are real enough, and writers like Clarke and Heinlein were usually pretty careful about getting things right. (Heinlein reputedly spent a couple of days doing orbit calculations - with pencil and paper - for a couple of lines in the YA novel Space Cadet.) We don't have atomic rockets, but one was ground-tested in the 1960s, at a place rather colorfully known as Jackass Flats.
Heinlein handwaved the Solar System quite a bit - by the 1950s hopes for canals on Mars or a jungle Venus had pretty much faded - but mostly what the SF writers of the 1950s got wrong was the cost. No fault of theirs; no one yet realized just how hideously expensive space rocketry would be, or how difficult it would be to spread out the costs by flying spacecraft on a regular basis. (747s are expensive too, but air travel is cheap because they can carry a full load of paying customers every day - for shorter flights several times a day.)
In one major respect, however, rocketpunk - and for that matter a good deal of space SF not intended as rocketpunk - invests in a great deal of negative handwavium. If the writers of the 50s greatly underestimated how expensive and complicated space travel would be, they even more greatly underestimated how cheap and convenient computers would be. A classic instance is Heinlein's YA Starman Jones - still one of my favorite SF novels.
Heinlein paints a vivid picture of life in a starliner's control room - including the petty officer whose job is to convert computer inputs into binary numbers, using a book of tables, then convert the computer's output from binary back into decimal for the astrogator to use. Forget scripting languages, forget C++, forget even FORTRAN or assembly code - a starship's control computer requires binary input, keyed in on a front panel. Hey, it was 1953.
The real problem for SF writers, though, isn't the guy looking up binary numbers, it's the astrogator. Even with current computer technology, the interstellar navigation problems portrayed in Starman Jones - the ship gets Lost in Space after its Astrogator dies - could surely be handled better by a computer than by weary Captain Blaine and incompetent Mr. Simes, or even by brash young Max Jones, the hero, or the late, brilliant Dr. Hendrix.
SF writers have resorted to various gimmicks to get around this problem. One fairly popular dodge is to say that computers don't work in hyperspace for some reason. That, however, is a painfully obvious bit of special pleading. (Well, so is FTL, but ....) Frank Herbert in Dune posited a distant future in which computers had been placed under religious interdict, which at least bellies up to the bar.
The need for this huge negative handwave, rolling back computer technology to the 1950s, is perhaps the biggest challenge facing rocketpunk as a subgenre. Rocketpunk isn't alone in that, however, because it is a problem that most space-oriented SF runs into sooner or later. Who would have imagined that one of our biggest literary problems would be too much technical progress?