Well, there are a number of gifts you can give Katey. What they and what I believe to be the most convenient locations to obtain them are as follows:
-Giant Stuffed Elephant - Stylin' Toddlers (Royal Flush Plaza)
-Giant Stuffed Donkey - Small Fry Duds (Royal Flush Plaza)
-Giant Stuffed Rabbit - Lombardi's (Food Court)
-Giant Stuffed Bull - Children's Castle (Royal Flush Plaza)
-Robot Bear - Astonishing Illusions (Royal Flush Plaza)
-Bag of Marbles - Ye Olde Toybox (Royal Flush Plaza)
-Watergun - Near the small circular stage (Fortune Park)
-Stick Pony - Ye Olde Toybox (Royal Flush Plaza)
-Funny Painting - Buy from Randolph during Art Appreciation.
-Beach Ball - Stylin' Toddlers (Royal Flush Plaza)
-Snowflake - Tame at the Yutacan Casino (see that achievement guide for more details)
You only need to give her one of these for the achievement. The closest and most convenient I believe is the Beach Ball, which is just across from the exit of the safe area behind the coffee shop. The Giant Elephant is also in the same shop, but is more difficult to carry.