According to Foodsafety.gov, fish may only be safely refrozen after partial thawing, meaning it retains some ice crystals and feels cold or measures below 40° Fahrenheit. If it has been fully thawed and held at more than 40° F for more than 2 hours, it should be discarded. WebMD recommends following the old adage, "when in doubt, throw it out."
While generally safe, refreezing partially thawed fish is likely to lead to some loss of texture and flavor, and is best avoided unless absolutely necessary. The same is true of some dairy products, such as milk and cheese. Partially thawed and refrozen fruits and vegetables are also liable to diminish in quality, but are generally safer than fish, provided they do not look or smell bad and they have not been held above 40° F for longer than 6 hours.