What’s special about a “rare” expedition

After some more investigation it looks like the names are affected by another skill the palico has.

• "Rath-of-Fast" has a "Team Attack Haste" ability (which increases setup time of team attacks)
• "Purrtuoso/Water" and "Rath-of-Water" have a "Water Synergy" ability (which adds an element to a team attack)
• "Rath-of-Signal" has a "Team Attack Signal" ability (which lets you manually activate the team attack by signalling to your palicos (Start+X))
• "Rath-of-Mix" one has both "Team Attack Boost" (adds "new properties" to team attacks) and "Dragon Synergy", so the "Mix" suffix probably just means the team attack is affected by multiple abilities.

So in other words just looking for their abilities that affect team attacks will explain its name!

The first stringers determine the skills your Main Palico will have. 2 First Stringers with the same Forte will give your main Palico the first tier ability of that Forte, 3 of the same type will give you the second tier ability of that forte, 4 of the same type will give you the third tier, and 5 of the same type will give your main Palico a special ability only he/she will have access to.

For example, suppose you have five first stringers, two Healting and three Combat. Your main palico would then have access to Health Horn, Demon Horn, and Piercing Boomerang. The Health Horn comes from the two Healing Palicos, and the Demon Horn and Piercing Boomerangs comes from the Combat Palicos.