Here are some items you can consider: - Thermometer - Antiseptic wipes - Gauze pads - Hand sanitizer - Disposable tweezers - Band-aids - Some medications and ointments for rashes and scratches List is not very comprehensive but I think it covers the basics... feel free to add on or comment! :)
For me, I always stand by calendula cream and mosquito repellent in my diaper bag. Calendula cream is used to apply on my kid when there is rash/mosquito bite. Mosquito repellent is always stand by to apply when we are going outdoor.
On top of what Jacq has listed, you can include calamine lotion for rashes, insect repellant to keep the bugs away and aloe vera gel to soothe sunburns.