M.P.
Yes, you can freeze it again if you've been handling the food in a safe way; i.e. it's not been left out for longer than it takes to eat and clear the food away. When food sits at a low temperature for an extended period of time food poisoning can develop.
The reason that it's often suggested to not refreeze food is that refreezing can alter the texture of the food which makes it less palatable but not dangerous. You'll have to experiment to find which foods your family still enjoy once refrozen and thawed.
The reason that the texture can change is that freezing causes cells to expand and in some foods break. Some foods become increasingly more mushy as more cells are frozen.
Later. J.L. is correct in that if the food is not handled in a safe way you are increasing the chance of food poisoning. You do have to refreeze it quickly once it's been heated. No letting it sit on the table while you're doing dishes sort of precaution. You can put it in the refrigerator and then refreeze it after you've done the dishes. The key is to not let it sit at a lukewarm temperature for more that a few minutes. I don't remember how long it is. I'll look it up later. I have refroze foods, usually food that has thawed and that I hadn't used, such as a TV dinner that thawed while I was sidetracked getting it home.
