Double reed players have lots of stuff. Reeds, swabs, water, tools, bocals, emotional baggage, etc. Here are a few of the things I use that just make oboe and bassoon life easier:
A water cup that attaches to a music stand. Water cups constantly spill when on the floor or balanced on the stand lip. You can purchase these or make your own (a binder clip, duct tape, and a prescription bottle work just fine). Tupperware makes a "midget" cup with a waterproof lid, and craft stores often have good lidded containers too.
A tuning app for the phone. I use Cleartune and Tonal Energy (this one also has a metronome).
Cork grease - I use the Vandoren brand, and it even smells good (fresh and citrusy). I got it on Amazon but it's in music stores too. Bassoons with threaded tenons instead of cork should use paraffin wax, and this is sold on the candle-making aisle of craft stores.
For oboes: A swab with a string on BOTH ends (a "rescue" tail). This prevents the swab from getting stuck, since you'll be able to pull it back out if it doesn't go all the way through. Note: DO NOT USE a clarinet swab for an oboe!
For bassoons: A swab that fits the boot joint AND the wing joint (separate swabs are ok but this is more convenient), as well as having a separate bocal swab or brush.
Cigarette paper (un-gummed) - Woodwind players get water in their keys (especially oboes, ALL the time), and cigarette paper sheets are traditionally used to get rid of it. Any type of absorbent tissue paper works though. Thin microfiber works too.
For oboes: A thumb rest cushion - not completely necessary, but does make playing a little more comfortable (due to both the padding AND the slightly lower thumb position). Clarinet thumb rests also work, and fun fact: so do cheap rubbery pencil grips.
A mini flathead screwdriver. No, your child should not be messing around with instrument screws, but it's nice to have one ready when a teacher needs it (I always have one on me).
A giant binder clip to keep school music stands from sinking. We all know how annoying this is. Just don't forget it when you leave class.
For groups outside of school, a music stand shelf (I use a "microphone stand shelf" from Amazon) is handy to hold all our stuff. I put my reeds, tools, pencil, water, and other things on mine during rehearsals and concerts. Not great for school though, since there's not much setup time.
For outdoor concerts or marching band: A necklace fan (O2 Cool makes a good one) that blows at your face while you play. Plan B: Just don't play outside.
For bassoons: 400-600 grit wet/dry sandpaper and needle-nose pliers for adjusting your reeds
For bassoons: A cup-style leather seat strap. The "S hook" type os ok too, but they can inconveniently come unhooked on their own sometimes.
Book of instrument jokes - for when the other musicians inevitably make fun of your instrument. Always good to have a comeback. Better to have these memorized though, or you'll be an even easier target.
Other random things I use that older students and adults might enjoy:
The biggest iPad Pro - good for organizing music for multiple groups without having to deal with folders and papers everywhere.
If going with paper music, use post-it tabs with the music titles so you're not thumbing through 50 pieces trying to find the right one.
A black folding fan for when concert halls are way too hot and we're required to wear long sleeves
A black make-up/travel bag to hold all my items on stage, hidden under my chair
For the ladies - a dress with POCKETS
A mini thermos to keep reed water warm - handy for winter/dry weather
Aquarium tubing from pet stores or airline tubing from Home Depot helps English horn reeds stay on the bocal (costs less than the stuff at double reed stores). Sometimes EH reeds already have tubing on them.
Nylon upholstery thread (for tying reeds) from craft stores is cheaper than "FF double reed" thread.
A solid black straight pocket knife - not an official "reed knife", but gets the job done and isn't distracting at concerts. Do NOT bring this to school (college is ok).
An extra-heavy oboe/EH stand (I have a "Jeanne" brand) that will never tip over, no matter what
A bassoon stand for when you have to also play all the oboe solos, lol