This is an updated version of guide made by Maulise here.
While all foods and toys have a small chance of giving stats, this is a guide to those items that have a higher chance of giving stats. No items have a verified percentage given, but a few have seemingly-accurate speculated figures - take these with a grain of salt.

Foods

Crocodile eggs - 1 use; 20% hunger, x% of 1 stat, 20 mood (February Event, Reptile Roundup prize [Flowering Fields Hard])

Egg Yolk - 1 use; 3% hunger, x% of up to 15 stats (high failure rate; usually gives 0-4) (April event, Whack a Snake prize, very low chance of getting it in April from bunny encounters, very low chance of getting in December from Bodybuilder Lion fight in Explore)

Fluffy Bunny Carcass - 1 use; 20% hunger, x% of 1 stat (February Event, April event, Reptile Roundup [Underground Tunnels])

Hippo Fat - 1 use; 20% hunger, x% chance of 1 stat, 10% mood (February Event)

Honeycomb - 2 uses; 15% hunger, x% chance of up to 6 stats, 10% mood (February Event, Slots Prize, Serengeti Shuffle prize)

Longfin Eel - 2 uses; 20% hunger, x% chance of up to 2 stats, 5% mood (February Event)

Small Turtle - 1 use; 10% hunger for adults, 30% hunger for cubs, 30% chance of 1 stat (World drop, Slots prize)

Turtle eggs - 2 uses; 20% hunger, 60% chance of 1 stat (February Event, Reptile Roundup prize [Flowering Fields hard], world drop in zone 2-5 level from AWD encounter)

Mudfish - 1 use, 8% hunger, x% chance for 0-5 stats (Can be found in explore in August Event)

Toys

Camel Toes - 1 use; x% chance of up to 2 stats (February Event)

Dry Palm Leaf - 5 uses; x% chance of up to 2 stats (February Event, December World Drop, Serengeti Shuffle prize)

Flamingo Legs - 3 uses; x % chance of up to 2 stats (February Event)

Kudu Horn - 3 uses; x% chance of up to 2 stats (February Event, May Arena Prize, Whack a Snake prize)

Lovebird Plumage - 2 uses; x% chance of up to 3 stats (February Event)

Ostrich Fluff - 3 uses; x% chance of 1 stat (February Event)

EXP items

UAV - Search and Rescue - 1 use, 20 EXP (August Event Encounter Drop, Environmental Explore Encounters)

Broken Drone - 1 use; 50 EXP (March World Drop, Wet Season World Drop, August Event, Slots, Serengeti Shuffle)

Coconut - 1 use; 50 EXP (August Event, Cups, Slots, Whack-a-Snake, Serengeti Shuffle)

Giant Tortoise - 1 use; 400 - 800 EXP (January Event, May Event, August Event, November Event, Gorilla Enclave, Whack-a-Snake, Serengeti Shuffle)