Is it safe to thaw chicken to have it brined and then freeze again?

In this case I would err on the side of safety and advise against refreezing the chicken for one reason - it was thawed in water for a couple of hours. Had it been thawed in the refrigerator it would be perfectly safe to do so.

It is well documented that it is safe to refreeze both raw and cooked foods as long as they have been kept out of the danger zone. With some foods, however, there may be a loss in terms of taste and texture.

People refreeze food all the time without even realizing it. Here, in the US, 'fresh' meats are most often shipped to grocery stores in a frozen state and then thawed in the meat case or thawed before placing in the case. Chicken is most definitely one of those foods. Consumers buy it 'fresh" out of the case and take it home and (re)freeze it.

From The New York Times :

The notion that you cannot refreeze thawed foods “is a myth,” said Tina Hanes, a registered dietitian with the United States Department of Agriculture’s Food Safety and Inspection Service. Any raw or cooked food that has been thawed can be refrozen as long as it was thawed properly — in the refrigerator, not on the counter — and hasn’t spoiled. That includes raw meat, poultry, fish and seafood, Ms. Hanes said.

“It’s one of the most popular questions we get on our hotline,” she said, “but it is safe to refreeze raw meat, as long as it’s not spoiled.”

In this case, I would suggest that you cook the extra chicken, cool it quickly, and then freeze it for later use.

I understand not wanting to waste the chicken that was thawed but as others have stated, once meat has been thoroughly thawed, it's simply not safe to re-freeze to thaw out later for eating even it's been soaking in brine. Since the chicken is raw, bacteria will be present. You know that cooked foods will go bad after a number of days even if kept in the fridge. Think of how much faster bacteria have been multiplying in raw chicken. The only thing to do is cook it now or throw it out.

I have no idea what you are planning to make with brined chicken but why not cook the extra portion (as is, not in the recipe you're making) and thenfreeze it? Since its now cooked, it's safe to freeze as any bacteria have been destroyed. You can use that frozen portion for another time, even if you can't use it the same way. There must be other ways to use it in meal preparation. Preferable to throwing it out.

If you thaw out any type of meat you must thoroughly cook the meat and let it cool before you freeze it again. Never, ever re-freeze raw meat.

Short answer is no because of something called cross contamination. Never ever re-freeze something that has been unfrozen. Have a wonderful day.