Chrono Trigger / Chrono Trigger DS
What are the changes? This GameFAQ tell us the story.
What is interesting is that you can choose two modes: Classic and DS. Classic tries to emulate the 16-bit goodness from the SNES, while DS Mode tries to put both screens to optimal use. This marks the first thing to make a remake a success: Add new stuff, but make sure not to disappoint fans by removing what was already classic.
This is further confirmed by the new, optional, areas. The first one is a Battle Arena, where you can battle CPU monsters and people via DS Wireless play. They actually added some kind of multiplayer there. The second one is a new sidequest with two new dungeons. The most interesting is the Dimentional Vortex, which grants access to three new dungeons and a new final boss.
Plus, you can carry it with you. Nothing better than playing a classic on the go.
Which brings us to the next point...
Secret of Mana / Secret of Mana iOS
Changes are mainly cosmetical. And, controlling the game with a touchscreen is a whole lot different than with a controller. According to this GameFAQ post, I only missed the fact that they fixed some bugs here and there.
What did we learn? Polish the graphics. What couldn't been done around 1990, can be achieved better in 2014. We learned one more thing: Fix those bugs. You might want to leave the glitches intact, otherwise you'll make your hardcore fans sad.
We'll take a somewhat weirder example next.
Toki Tori GBC / Toki Tori WiiWare
What we see here are fundamental gameplay changers. In the Wiiware version is the time limit removed. Puzzle games don't need time limits. The running feature is gone, and is replaced by a feature where you can point to the place where Toki has to go. The new version also has a Wildcard where you can skip just one level with. The levels are also different. In the versions after Wiiware (e.g. Steam) there was also a rewind function added. These versions also got extra levels based on Valve's smash hit Portal 2, combined with an ARG on the Steam version.
Conclusion? Make the gameplay modern. Just making AI smarter (like in Secret of Mana iOS) is good, using easier controls (point to walk) or make sure casual gamers can beat the game too (á la New Super Mario Bros. Wii's Super Guide, or even an easier difficulty) or make the gameplay less frustrating (rewind feature).
Question for you: how do you remake the game, that millions of players tagged 'Best Game Ever', loved by the whole gaming community, the game that lots of players are still playing nowadays? That's right, we're talking about...
The Legend of Zelda: Ocarina of Time / The Legend of Zelda: Ocarina of Time 3D
First one is the Sheika Stone. Once you get inside, you get help about what you exactly need to do. It's a bit oddly located: there is one at your tree house at the start of the game, and one at the Temple of Time. You can also get to it when your hearts are depleted.
Second is a recycled game mode from another port from the game. The dungeons are remixed: nothing is what is familiar with you after beating the game tons of times. In addition to that, Hyrule is mirrored. Nice service for the ones who beated the game over and over.
Lesson? You don't need to change that much. OoT has now become portable, Hyrule was given a fresh coat of (3D) paint, and there are some new things here and there. I personally think it plays a lot better because of the Gyroscope aiming for some weapons, and the better Circle Pad. Aiming in the N64 version was just pretty bad :(
Now I really need to play Chrono Trigger! All games and trademarks are owned by (in order) SQUARE, SQUARE ENIX, Two Tribes and Nintendo. Screenshots and videos kindly taken from the internet.