If your son is not having an injection with the snack (and the snack is not a required snack because he's on a mixed insulin) then his readings will go up with any food that contains carbohydrate. You're not looking at just sugars, you want to count the carbs as they are what affects blood sugar. So flapjacks for example might be low sugar but the oats are high in carbs which is what will be causing his blood sugar to spike. The other foods suggested by Tracey69 will also make his blood sugars go up if he isn't taking an injection to compensate the carbs, they are all high carb foods. If he wants to snack on something that doesn't contain carbs so that he can avoid an injection to cover the carbs then here are some examples:
sugar free jelly
eggs
cold meats (chicken drumsticks, ham)
cheese
celery, cucumber, carrot sticks with a dip like gucamole
avocados
mushrooms
salads with meat (ham, chicken, bacon), eggs or fish (tuna)
tomatoes
These following snacks are low in carbs but not carb free. sometimes you can get away with a low carb snack without an injection but you'll need to test to find out:
berries (strawberries, blueberries, raspberries)
melon
plain lightly salted popcorn (small portion)
natural yoghurt
Is your son on a basal/bolus insulin or a two injections a day mixed insulin? If basal/bolus then he can have a sandwich for example but would need to inject the right amount of insulin to cover the carbs in that snack.