Why does the LED on my son's delay switch stay on all the time?

The most common cause of 'LED comes on and never goes off' in these delay circuits is that the delay is just much longer than you think. If the book used a big capacitor with a big resistor, the delay can run for many minutes or even longer, and it feels like it never shuts off. An easy test: temporarily swap the delay resistor for a much smaller one, or put your finger across it, and see if the LED turns off a lot sooner. If it does, the circuit is working and the parts are just sized for a longer delay than expected. Second thing to check is the big round timing capacitor, if it's the electrolytic type with a stripe down one side. Those are polarized, meaning they only work one way round, and backwards they behave realy strangely, sometimes holding a charge forever or never charging properly. The stripe should go to the more negative side according to the book diagram. This is probly the single most common mistake on a first breadboard build, and it's invisible from above becuase the part still fits in the holes. Third, test the sensor itself. Cover the light sensor completely with your hand in a dim room and watch what happens over a minute or two. If nothing changes at all, the problem is upstream of the timing parts, in how the sensor connects to the transistor. Also double check the transistor's leg order against the diagram, since the flat side faces a specific way and kids (and adults) get this wrong constantly.

One practical method that worked well with my kid: sit down with the book and have him point at each component on the diagram, then find that same part on the board and say what he thinks it does. You'll usally find the mismatch within ten minutes, and he learns more doing that than from you fixing it. A cheap multimeter helps a lot here too, just checking wether voltage appears at the capacitor and then drains away tells you if the timing part is doing its job.

Still finding my way around here so sorry if this is obvious, but when I helped my nephew with a similar build we spent ages checking parts before realizing the issue was the breadboard itself. Some rows just didn't connect, probably a worn board. Wiggling the jumper wires changed the behavior each time. Might be worth having him gently press every wire into its hole again before touching any components. Does the LED behave differently if he taps the board lightly? That was the clue for us.

When my daughter built one of these I photographed the breadboard from two angles before touching anything, then compared it against the diagram at dinner. Turns out she'd shifted one resistor three holes over and everything else was right. Photos make it way easier to spot a single misplaced leg than staring at the board in person. Also, with the dark-sensing behavior, try pointing a flashlight straight at the sensor and see if the LED reacts at all. If it flickers when the light hits it, the front half of the circuit is alive and your problem really is just the timing side. If it's totally dead either way, work backwards from the sensor instead.

Update: took the photo idea and compared his board against the diagram at the kitchen table. Everything's in the right hole as far as I can tell. But that first reply got me thinking about the timing. We swapped the delay capacitor for a smaller one from the spare bag and now the LED shuts off after a few seconds. Was it really just that the original setup had a delay way longer than we were waiting for?