One common problem I see as a teacher is when a child has chosen an instrument that doesn't suit them. These students typically get frustrated quickly and quit after a few months. When a child chooses the wrong instrument, it can be a negative experience for them and they may have fears about trying a different instrument in the future.
Many people think it doesn't matter which instrument their child plays because they're all the same, but that isn't true. Children have very sensitive hearing, more so than adults, and they will quickly gravitate towards and away from certain sounds. The first step is to have your child listen to various recordings of bands and orchestras, iTunes and YouTube are great places to start for this. Simply searching for videos using terms such as symphony orchestra, symphony band, jazz band, or marching band will bring thousands of results. Live performances work even better, check to see if there's a local orchestra or band in your area. They will quickly begin to narrow down which instruments or instrument family (ex. brass, woodwinds, strings, keyboard, or percussion) they like.
Once they've got it narrowed down to a couple of instruments or to an instrument family, have them watch videos of soloists playing these instruments. YouTube works perfectly for this and is as simple as typing the name of the instrument into the search bar. This will show your child what each instrument is capable of. They will also discover whether they like the look of the instrument being played as well as understanding the different sounds of instruments in the same family. This means the difference between the sound a french horn and a baritone, violin and viola, etc.
Now that they are pretty sure of which instrument, contact a local music store and bring them in to try out this instrument. This is to help them discover if they like the look and feel of the instrument in person. They will quickly decide whether the instrument is going to be too heavy or cumbersome for them to hold.
The last step is to have them take a trial lesson with a private teacher. The child will learn how to blow into the instrument and make a sound. They'll also learn the basic set up and maintenance for the instrument. Flutes have fairly easy set up and maintenance, but are the hardest to get a sound out of. Brass instruments on the other hand can make a sound quickly, but have to be oiled and greased regularly.
I know this sounds like a lot of work just to get your child set up with an instrument, but it will save time and money down the road, especially for the child that wants to switch instruments every two months. The last thing is that parents should be objective during this process. I've seen many children that are unhappy with an instrument simply because their parent picked it for them.