You will need some watermelon, a blender or food-processor, and some popsicle moulds. This is especially good to do with over-ripe watermelon that isn't as nice to eat as is.
Slice the watermelon, cut off the white and rind, then cut the watermelon into 1" pieces. Remove any seeds. Put the melon into your blender or food-processor and blend until very smooth. Pour into the moulds, and freeze. Easy as that! Any leftover blended watermelon is nice to drink.
Some other things that make tasty popsicles are unsweetened applesauce mixed half-and-half with unsweetened fruit juices (the applesauce keeps the popsicles from getting too hard for kids to eat easily). You can also freeze most fruit smoothies (although smoothies with yogurt may give you a different texture). You can try blending other soft fruits: first remove skins from fruits like peaches, and you may need to add a little sweetener to tart fruits. As a kid, I loved raspberry popsicles, made from just mashed raspberries with a bit of sugar (and these days you have so many options for alternative natural sweeteners).
I hope you have fun plans for the Summer to come!