Gonna answer in 2 parts.
1. The easiest way would be to just Google what to do for each Rare Blade quest because otherwise it's random as hell. In most cases you need them in your party
and need to visit certain areas. So unless you know who to use and where to take them it's pure luck on whether or not the quest will start.
2. Each Blade (and character) has two favorite Pouch categories and items....for example, Rex's is Drinks and Art. Specifically Pan-Fried Tatari and Coral Reversi.
An item's effect is multipled by 1.2 if it is in a favorite category of the Driver or Blade, 1.5 if it is a favorite item, or 1.8 if it is both.
More importantly, repeatedly using a Favorite item will increase Trust for a Blade, which makes Affinity increase. Some Blades need specific Favorite items to open up parts of their Affinity chart....for example, using Poppi's Favorite items will increase her Jump Field Skill.
Here's a list of Favorite categories/items if you want to "cheat"....otherwise it's a lot of trial and error:
Pouch
Fred.