I bring a 100ml squeezy bottle (from Muji's travel bottles selection) of washing up liquid to last me a week rather than buying a bottle out there.
I also bring a box of 40 teabags, as I like a certain brand and can't get them abroad.
Robinsons (and other brands may also do similar) have a very concentrated squash in a very small bottle that can be handy to bring and makes water more interesting - make it using sparkling water and it's lovely! If you bring a water bottle each as well, you can fill these up daily before going out and that will save on needing to stop at shops/cafes for every drink - but you will still need some extra drinks I reckon.
If you have them, bring a small salt and pepper, and maybe 1 or 2 favourites herbs/spices/seasonings. If you can find them, small medicine cartons or little facecream pots (empty and clean!) might work well as you only want a small amount of each. But might allow you to make something like a frittata omlette easily, or whatever your family likes.
And maybe a small amount of a wine vinegar in another travel bottle or similar - 50ml or less should be plenty. For over chips if needed, to make salad dressing (with olive oil below, and your herb above) etc
I normally throw in a J cloth or similar, as I never know what the facilities will be in SC and it takes up almost no space.
Food I would buy on arrival (in a reasonable grocery, not the tiny ones on-site, if possible) are:
a small bottle of olive oil (250ml)
real butter (if your family uses spread, get that, but I like the treat on holidays and it can be used for cooking as well)
block of cheese
couple of onions and a bulb of garlic
eggs
some nice veg and salad - lettuce, tomatoes, peppers, mushrooms and other bits that look good
some nice fruit
lunch things - cooked meat, quiche, ...what your family will eat, check out the deli counter for nice or different things to try
enough for 2 dinners - easy ones, but nice fresh ingredients
snacks - you are on holiday - fruit is included and cheese, but also some crackers, biscuits, crisps, nuts, olives etc - again what your family would like and eat
breakfast - cereal if needed, OJ/apple juice, bread, jam, croissants etc
coffee, hot chocolate? (I bring the tea from home)
milk
something else to drink - beer, wine, sparkling water, still water, fizzy drinks etc
Kitchen towel (can use sheets as napkins too) and perhaps some extra toilet roll