Siegal, your original post actually raises two issues.
First, the general issue of refreezing previously frozen foods. Assuming proper defrosting (which means in the fridge), what you have with refreezing is more a loss of quality problem than a food safety one. Usually has to do with cell structure and ice crystals and such. If you use that product as an ingredient for a cooked dish, however, then you are, in effect, freezing it for the first time. There should be neither a quality loss nor a food safety issue.
When it comes to liquids things are slightly different. If you were to properly defrost some stock, then immediately refreeze it, there would be no loss of quality, because there is no cellular structure to be damaged.
The issue with stocks, however, is that unfrozen they lose quality very quickly, and, even under refrigeration bacterial growth can cause problems. Rule of thumb is three days in the fridge, maximum. After that you should either dispose of it, or reheat it to the simmer point, let it cook at least ten minutes, then refreeze.
To avoid the problem, I freeze my stocks in amounts ranging from one cup to one gallon. Then I try and defrost only the amount I'll be needing. F'rinstance, I might just defrost one cup for a sauce, but go for a gallon if I'll be making soup.
Naturally, it doesn't always work out that way. But I try to never have more on hand than I'll use in the next few days.
One other thing about stocks, that many people ignore. They should be cooled as quickly as possible. Otherwise they'll spend far too much time in the danger zone (40-140F). To avoid that, put the stock pot in a tub of ice, and stir it more or less constantly until it's cool enough to put in the fridge.
Most people spoil garden things by over-boiling them... if they are overboiled they have neither any sweetness or beauty. Hannah Glasse 1745