Instructions
Evolve Togepi. As with any other Pokemon, this means you must use Togepi in battle. However, your objective is to raise its happiness, not its experience. When its happiness bar is maximized, give Togepi a Shiny Stone to hold, purchasable in the Poke Mart for 10,000 Poke Dollars. This causes Togepi to evolve into Togekiss.
Go to the daycare just south of Goldenrod City on Route 34. Give your Togepi and a Pokemon of the opposite gender to the woman.
Leave the daycare and walk around for 15 minutes real time. Each step you take in the game is another minute that goes by, and the longer it takes for you to return, the more likely it is that the two Pokemon have bred.
Return to the daycare and speak to the woman. She will give you an egg, which you must keep in your inventory while you go about your journey. Leave the daycare and take 1,000 steps in-game, causing the egg to hatch.
Repeat steps 1 through 4 if your Togepi is not a female.