Well, some of it is what "expectations" are. And some of it is cost. And some of it is the desire of the student and also what the parents are willing to "put up with".
Many of the woodwinds end up being girls. The flutes, clarinets, oboes, etc, are almost always exclusively girls. There is no rule that says it must be so, but generally, that is how it shakes out. Most of the brass ends up being boys--again, there is no rule, and you will have some girls in the brass, but the majority will be boys. These are the tubas, the trumpets or cornets, the french horn, the trombone, etc.
Saxophones are actually woodwind instruments. But, typically, for whatever reason (size maybe?) there are a lot of boys who play them. Both the alto and the tenor sax.
By and large the brass instruments cost more than the woodwind instruments do. But pretty much any instrument will be available through a rental program (usually somehow through the school or the school will subcontract out with a particular music store), but the prices for rental are going to follow the cost of the instrument.
Both boys and girls play percussion. Which isn't JUST drums. But a snare drum is going to be a basic fundamental part of it. Drums aren't cheap either. And they are loud. Most schools have their own drum sets that they keep at the school for the kids to use at school. Providing something at home to practice on is something you have to check out space/cost wise. All the other instruments the student will have to carry back and forth between home and school, so that they can practice daily at home and also have it at school each day during class time.
It isn't nearly as much of a hassle as it sounds. I promise. It is just something that becomes part of "what they do". But it is a reality, and if your child rides a bus, it is something to consider when he is asking to play the tuba instead of the trumpet.
If your child isn't happy, he can ALWAYS change the instrument.
Most of the time, schools will have an open house type 'event' where the students can come in and give several different instruments "a try". If not, call a local music/instrument store and ask them what they can offer/suggest to you in the way of testing out what he likes. I'm sure they've heard the question before.
Good luck. Music is a wonderful addition to any education. :)
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ETA: For those calling out my references to what sex plays what instrument, get a grip. I did not say anything that is not the truth. In MOST cases, that is how it shakes out. I explicitly stated that THERE IS NO RULE, but what I said is TYPICAL. My own SIL played just about every brass instrument there is. We had A boy play the oboe in my high school band. And one boy played the clarinet. Out of about 30 clarinets. For some kids, it makes absolutely no difference whatsoever. But for other kids, it is a HUGE deal for them. I don't know the original poster, or her child. But it may be part of what they will want to consider depending upon the child's personality. For example, if I had a son who was constantly getting teased and had been a target for bullies in the past or something, I would NOT encourage him to play an instrument that IN MOST CASES is played by girls in the school bands (like the flute). It would just be another glaring thing that other kids COULD use to be mean to him. I would want to know what is typical, and what might be atypical. That is ALL I was pointing out. Geesh!