After thorough testing, I can say with quite some certainty that I understand how Inver's rewards work. Here are my results:
The number of items you receive is 3 if none of your Pokémon faints, 1 otherwise. I've heard rumors about getting 3 if you lose, haven't tested this, but assume it's a hoax.
The item depends on the number of super-effective (S) and not-very-effective (N) hits you land during the entire battle. The difference S-N becomes your number which determines the item category. Within a specific range of numbers, which item you receive is random. Moves with normal damage have no effect on this.
You have to start before midnight for it to count for that day. How long the battle takes doesn't matter, you can still battle him again if the date rolls over during the battle.
Changing time or date on the 3DS resets the timer and you will have to wait for 24 hours until you can challenge him again, possibly more (it's possible the actual time is 24 hours + whatever time it takes for the date to roll over after that). Changing it back won't fix this.
Here's the list of Item ranges:
0: Oran Berry
1-3: Sitrus Berry
4-6: Type-resisting Berry
7-9: Evolutionary Stone (includes Everstone)
10+: Rare Candy
And here's my list of Items received on each number, starting with the Berries (which are noted by the type they make the carrier resist), followed by the Stones. This is the data I had available to reach the above conclusions (* denotes results since I found out the formula is S-N):
4: Dark, Flying, Psychic, Rock, Normal*
5: Poison, Grass, Fighting, Ghost*, Normal*
6: Fairy, Dragon, Steel*
7: Sun*, Leaf*
8: Moon, Fire*, Moon*
9: Dawn, Shiny, Everstone, Everstone*, Leaf*
One last thing I have for you is the ultimate Inver farming strategy, which is the way I got all this data and as far as I can tell the only way to reliably reach anything above 6 without having some of your own Pokémon faint. I'll do a step by step guide.
- Get an Anorith. Ideally from someone else (Friends, Wonder Trade, GTS).
- EV train it for bulk. I went with 252 HP and 128 each of Def and SpD. Use either conventional ways or Super Training, it doesn't really matter. But you should do this before anything else.
- Train Armaldo to at least Level 75. As he's in the Erratic Experience group, this should go rather quickly. The best way to do that is to give him Lucky Egg and hunt down Furisode Girls in the Battle Chateau on Route 7. I can add a detailed guide for this on request.
- Use TM54 False Swipe on your Armaldo. Give him Leftovers.
- Save in front of Inver.
- Battle him, doing as many S hits as possible. Count them (and the N hits)! Wait for a Ghost, Rock or Steel-type to appear.
- If he has no Pokémon with at least one of those types, reset and try again.
- Once the right target is there, False Swipe it until you have the right number of S hits. (You should target the central number, for example 8 if you want a Stone.)
- Defeat all of his Pokémon without doing any more S or N hits. A balanced (and strong or at least bulky) team makes this easier. Damaging Status (Poison, Burn, situationally Hail or Sandstorm) also helps.
- If you hit super-effectively too often, get in a corresponding number of N hits.
- Collect your prize.
Don't forget to heal your Pokémon so none faints. If something goes wrong, just reset and try again.
This way you can get 3 Evolutionary Stones or Rare Candies per day, depending on your needs. Note that if you're going for Candies, you can do as many S hits as you like beyond the 10th, so Step 9 and 10 become obsolete.