The daily gift is the Quake EP ability, an ability that does fixed damage to all enemies despite resistances. " What Rough Beast Slouches " can also be attempted if you are so inclined.
At the end of the day, the game will calculate if you have done enough to acquire an extra day. If so, Yggdrasil will gain another bloom to represent that extra day. if you have completed almost all side quests, you will easily have enough.
The Extra Day: Completing the five Main Quests will ensure you reach The Final Day and enable you to complete the game. However, if you have completed a sufficient number of side quests (around 37-40 which is all but a handful. Canvas quests do not count) you will gain enough Eradia to buy another day for humanity. On this day the Ultimate Lair is open for business, enabling you to defeat all Last Ones and reap the rewards. There is a catch though; enemies will grow in strength twice. Once on the extra day and once and the Final Day. You must therefore, make improvements to your Schemata to compensate.