You should be able to do quite a bit on the heat capacity of water using just an electric kettle and a cheap digital thermometer. (A cheap plastic jug kettle is probably best, and a 0-100 thermometer not a clinical one).
Half-fill a kettle with measured amount of fridge-cold water, measure temperature at start, turn on electricity for maybe ten seconds (timed), turn off, swirl to mix water, wait a bit, swirl again, measure temperature. Repeat several times until water too hot to handle safely. Plot graph.
To a fairly good approximation, energy in and temperature rise are proportional. It is much more challenging to determine to what extent this is not true, and whether this is due to heat leaking in or out as the water gets hotter than room temperature, or due to the water itself changing as it gets hotter. Both are true.
Study of experimental errors. Repeat with more cold water. How repeatable are your observations. What can make them better? How accurately can a human being with a watch time ten seconds of "on"? (I once read in a book, 0.2s - BS! I'm a musician. Getting fingers round ten notes per second is a major challenge but it's easy to hear if something is a tenth of a second off the beat -- so you should be able to know if the switch clicks are early or late compared to the watch ticks, and target 0.05s sigma? Digression, or another home experiment?)
So, you just might might be able to spot the changes in the heat capacity of water at different temperatures. IIRC about 0.75% change between 5C and 40C, most marked at the cold end. If you can start with water at 1C it's a somewhat bigger percentage change. Not easy to spot with such simple apparatus, but maybe just possible.
Obligatory safety note: supervise your child (hot water is dangerous, but explain the danger, and let him swirl cold or luke-warm kettles which can't do any more harm than wet clothes and floor. Knowing how to do experiments safely is important. I once came closer to killing myself than I like to recall, now the lesson has sunk in.)
And stop if your child is getting bored.
BTW it's not hard to build a continuous-flow liquid calorimiter at home and nail down the varying heat capacity of water, but that'd probably a bit too complicated for now.