I love this topic, and it's been a big deal in my life for years. First single with sometimes random "nightly plans" and then married with kids with random "day events"
Here's some stuff:
for first aid, we ahve a few bandaids, but mostyly a roll of 3/4 inch waterproof medical tape, gauze, and a half dozen butterflies.
Iodine. Iodine is awesome, iodine is great. It's absorbed readily through the skin if you are deficient or worried about fallout, but it's REALLY awesome because it is great for wounds AND purifying water.
a bit of powdered, buffered, vitamin C (multiple uses)
benadryl dissolvotapethingies (easier to get kids to take than pills or syrup, easy to carry, look like breath strips)
2 pr latex gloves.
one swiss officer in the first aid kit.
what else do you need? well, 2 led flashlights, one keychain, one bigger. some tape (I carry duct and electrical), a leatherman crunch and a cybertool are both in my urban assault bag, along with a few specific tools for my daily ride bike (those come out if I'm not on the bike)
napkins. I cannot believe no one has mentioned this. Ever gone into a toilet in a public space and been confronted with the no tp thing? or just had a pair of muddy hands from rescuing pocket change? Check oil or other fluids in a vehicle? napkins are indispensible, free, and disposable. Long before children came along, I always carried some, and they are always extras from a food establishment or cafe.
Book. (these days that's electronic books on my treo, which has a AA cell charger)- having something to read is really great for a lot of urban "stress" situations where you just have to wait.
food- always enough stuff to make a scratch meal for the family, often enough in calorie terms for 2. includes a couple chewable vitamins.
any meds I or the kids might be on, anywhere from the full scrip down to 2 days worth, depends on how bulky it is (it's easier to carry 30 loratadine than 2 because 2 will get lost)
bandanas! similar to the washcloth, I generally have 2 on me.
water. one bottle for each person, generally a nalgene. (yes, we even have a nalgene SIPPY cup for the 1 year old. It was on sale, and cute)
diaper- well, what do you expect, we have a couple diapers on us all the time, with wipes. Aside from kids, the wipes are REALLY handy to have around. Same with maxipads, which I've used to bandage a couple nice heavy flow wounds. Don't recommend used ones for this purpose, of course.
other stuff, this isn't a complete list. for example, my wife and I both have ham licenses and carry dadios, we have ponchos, usually have spare clothing for the kids, etc.
one other note is that since I added a "4 year old happy kit" into my son's backpack (honey, salt for his hardboiled eggs, plastic spork, bandaids and alcohol wipes, orange bandana) he takes the pack EVERYWHERE. It's stuff he can safely use, it's cheap, and it gets him into the habit of USING a kit