I'm thinking about having my daughter acquire "absolute pitch" ("AP"). AP (aka "perfect pitch") is the ability to identify or re-create a given musical note without the benefit of an external reference. More plainly, if you have it, you can tell which sound it is in the musical scale ("C, D, E, F... " or "do, re, mi, fa...") when you hear whatever sound it is. It's an amazing ability. Even when you hear a chirping bird or a dripping tap, you can identify its note. That well-known classical musician Mozart is also said to have had such an ability (he was said to have the ability to SEE sounds as various colors!).
It is said that children can acquire this ability only during a limited period of time (between 2 and 6 years old) and that it generally takes 1 to 2 years to earn it. The earlier you start, the easier. Once you earn it, it will last for the rest of your life. There are some music schools where they train children for AP in Japan (mainly in Tokyo).
In Japan, not a few parents at least once consider the same thing as myself especially if their children are females. The reasons are sometimes unclear. In most cases, it is not that those parents want their children to become professional musicians. In fact, they tend to think the ability as a gift for their children only parents can give since its window is open only for a limited time when they are too little to decide anything on their own. They expect their children to appreciate it in the future because it would make their lives more joyful. These children will be able to compose and play music very easily.
From time to time I clumsily strum my guitar and hum a song for fun. I imagine what my life would be like if my parents had let me acquire the ability...