The reasons for remaking a good game tend to be quite different from remaking a bad game. For example, Resident Evil was remade on the GameCube because the limitations of time and the hardware meant that a lot of content had to be cut from the original on the Playstation. One of the main reasons for remaking it was to tell the whole story that they couldn't before. To use Durion's example of the Pokémon games, the remakes served a few purposes but one of the more notable ones was making generations of Pokémon available in the later games (this is why I personally think Gen 3 remakes are not coming anytime soon; they just aren't necessary for anything). Then you get unecessary remakes like the recently released Fable: Anniversary, or Halo: Combat Evolved Anniversary while we're at it, which offer absolutely nothing except redone graphics. There is little to no point in remaking these games since there were no changes to gameplay, design, or story. The remakes of Ocarina of Time and The Wind Waker can be seen as another example of this, but at least they make use of the new hardware to make the gameplay experience a much more intuitive and, therefore, enjoyable one. Still a bit pointless but they try to justify themselves more than not at all.
Remakes of bad games are, as far as I have seen at least, received with a mixed response. Tell peope that Great Game You Loved is being remade and they usually get excited because it can be no worse than the original, right? But tell them Bad Game That Was Bad is being remade and you find people who question why such a bad game would get a remake, some people who say that the game was never that bad in the first place, and then others who say that a remake can or will fix all of the problems and make Bad Game into a Good Game.
While not really a remake, Hitman: Contracts does feature several missions from the original game, re-worked and re-designed to make use of all the gameplay advances made in the second game. The original Hitman: Codename 47 is, to be blunt, a bit crap. It's fiddly, clunky, poorly designed, and lacks the real freedom for choice that is a trademark of the later games. Contracts takes, as I say, several missions from that game and remakes them with that sandbox gameplay. The effect is transformative, turning some strict and, at times, tiresome missions into some of the series' most open and rewarding. The problems of the original were fixed by the advances in the remake, so I would say Contracts is an example of a bad game benefitting from a remake, even it is is only a partial one.
As for bad games I feel could benefit from a remake Far Cry 2 springs to mind. I know a lot of people liked that game but I thought it was quite rubbish but only because of a number of fairly small things. A remake, I feel, could improve that game by: reducing the frequency with which guns jam and deteriorate; give you more malaria pills, or even just get rid of malaria; offer more bus stations so getting around isn't as much of a pain; make it shorter since the landscape and mission types are not varied enough to be interesting for even half the length of the game; make stealth more effective, since it really isn't accomodated for very well with regard to weapons, equipment, or enemy AI. However, fiddling with the game in such a small way would feel more like a patched version than a remake and so would probably lack any real appeal, even to the game's fans.
Another would perhaps be Resident Evil: Survivor. Survivor was the first Resident Evil to use a first-person perspective. It was a light gun game that wasn't compatible with a light gun in North America. It was also the first Resident Evil to use all 3D environments and it really made you understand why the previous games, and some of the later ones, used pre-rendered backrounds instead. The extremely low draw-distance, combined with the first-person view, didn't help any either. Survivor was awful to the point of hilarity, I recommend you play it if you ever get the chance. Despite all of its many, many, graphical and gameplay failings, Survivor did tell an interesting part fo the Resident Evil story, following a man named Ark Thompson as he investigates the appearance of B.O.Ws on Sheena Island. Long story short, turns out Sheena Island is where Umbrella are mass producing their Tyrants. While the fine details of the story could mostly be ignored, Survivor gives us a clear insight into the way in which Umbrella works and highlights their threat to global security better than any of the previous, or even later games, do. The game could benefit from a full remake, one which even changed most of the story but kept the key points of mass produced Tyrants and teaching us about Umbrella. While the more action oriented design of Survivor would fit in extremely well with the modern 'dramatic horror' approach of Revelations or Resident vil 6, it does provide Capcom with a good excuse to revive the classic gameplay style, with necessary tweaks to make it relevant to a modern audience, of course. A Survivor remake that played with fixed cameras, limited resources, and tried to recapture the slower paced horror of the early games could prove to be a fairly popular thing, or at least would generate a lot of attention and get people excited about Resident Evil again. It'll never happen, but I think it's worth having.
