In terms of making a great remaster it should give the user ways to experience the Initial release in all of its intentions while showcasing enhancements that build off of that foundation in a newer version. These can be done in various ways
1. Older games tend to have graphics that may of aged poorly over time. Adding improvements to the way a game looks with textures can breath new life into the experience.
2. Controls for releases at the time may be harder to maneuver or make it less enjoyable to handle. Changing how a person can move through the game can make it less of a hassle to explore the world.
3. Original soundtracks and music can feel dated or have flaws. Improvements in how sound is produced or re-recording previous works can immerse you into every place you explore. This makes you feel like you live in that place.
4. Companies or teams don't always catch bug or glitches before release the first time and this can result in poor performance or issues the make the game unplayable. This gives time to the team to fix what was wrong so there are no issues.
They should have these aspects but there has to be respect for the original. If you change too much or alter the story too far then does it really feel like a remaster or a totally different experience?