I reckon that a lot of these games, especially the 8 bit games, would either be unrecognisable or feel terribly limited if they were remade.
As an example, when I got my xBox 360 I downloaded an updated version of R-Type from Arcade. In the late 80s it was the pinnacle of sideways scrolling shoot ’em ups. The xBox version had polished graphics but retained the original gameplay. And it was pretty disappointing. The gameplay relied far too much on learning the attack patterns of each enemy wave and was obviously oriented around repeating each level over and over. It felt really, well, dated.
That said, I think that a new Paradroid could be made, substantially different from the original (e.g., in 3D, possessing a real plot) but still feel related to the original. There’s more in the original concept that can be retained in a new game than there could in Uridium (its scrolling shoot ’em up sequel, mentioned by the OP I think).
I’d also like to see an FPS that taps into the spirit of the original doom. Frantic action, huge numbers of enemies attacking at once, lots of guns. Too many FPSes today attempt pseudo-realism, annoying sneaky missions, needing to reload guns, ordering squadmates about, etc., rather than just frantic shooting action.