From Bulbapedia, the community-driven Pokémon encyclopedia.
The Pokémon Center Ladies are a trio of non-player characters that appear in Pokémon Emerald. They are all located in the Lilycove City Pokémon Center, however only one will ever appear in a player's game. The lady that appears is determined by the ID number of the game.
Each lady will ask for something of the player. Once this is accomplished, she may give a reward.
The Contest Lady will ask for a Pokéblock to give to her Pokémon. The flavor of the Pokéblock needs to fit her Pokémon's preference. She says thanks but doesn't appear to alter the game in any way.
The Favor Lady will ask for an item. In return, she will give a Vitamin as a prize. However, she is cryptic and will only hint at the kind of item she wants using a description of what it looks like.
| Something pointy
|| Poison Barb, Sharp Beak, Durin Berry, Rabuta Berry, Tamato Berry, Spelon Berry, Stardust, Star Piece, Quick Claw
|| HP Up
| Something slippery
|| Repel, Super Repel, Max Repel, Parlyz Heal, Burn Heal, Antidote, Pecha Berry, Grepa Berry, Belue Berry, Fluffy Tail, Awakening
| Potion, Fresh Water, Soda Pop, Lemonade, Light Ball, Hard Stone, Lava Cookie, Chesto Berry, Kelpsy Berry, Nomel Berry, Durin Berry, Wepear Berry
| Something roundish
|| Rare Candy, Pearl, Big Pearl, Max Repel, Smoke Ball, Revive, Max Revive, Hard Stone, Shoal Shell, TinyMushroom, Big Mushroom, Aspear Berry, Oran Berry, Magost Berry, Watmel Berry, Grepa Berry, Poké Ball
| Something shiny
|| X Speed, X Attack, X Defend, Blue Flute, Yellow Flute, Black Flute, Red Flute, Sun Stone, Stardust, Star Piece, Pearl, TwistedSpoon, Premier Ball, Luxury Ball, Heal Powder, Nugget
| Something sticky
|| Energy Root, Max Potion, Full Restore, X Accuracy, Guard Spec., Dire Hit, Watmel Berry, TinyMushroom, Rare Candy, Leftovers
Although there are many other items that also fit the descriptions the Favor Lady asks for, she will only accept those listed above. If a wrong item is given to her, she will not accept another until the next day.
The Quiz Lady only appears after mixing records with a friend. She will ask the player a question; if it is answered correctly, he or she will receive an item.
The Quiz Lady will also offer the player the chance to create their own quiz using nine words or phrases. In this event, the player will also select an item from his or her Bag as the prize to those who answer the quiz correctly.