Hi, Leyla,
You're definitely thinking about the right things. Piano is great, but it can be lonely. The piano competitions are great for developing skills, focusing, performing, and dealing with nerves, and a win always has resume value. But my dc (who has kept up with piano, including competitions) really enjoys playing an orchestra instrument too for the social aspect. There are also different musical conventions: there's less memorization, so it's less stressful on that front too.
Perhaps people have different suggestions for instruments, but here's our experience. Bassoon and French horn are always in demand, and oboe generally too. These are all hard instruments, but they're rewarding. Oboes are expensive (a good one is $11K new, $8K used). You won't have to spend that till your dc is serious, though. Oboe study at a high level requires reed-making (bassoon too), which is essentially a woodworking skill, very difficult.
Brass players generally are in demand in youth orchestras -- so not just French horn but trumpet, tuba, etc.
