Hi,
I don’t know if the cough is a sign of anything more serious or if it is just a related issue of having throat surgery followed up by radioiodine therapy, however some things that help coughs might help bring some relief. It is very important to keep your throat moist by drinking plenty of liquids. Hot tea, water, honey, lozengers and hard candies can help sooth the throat and might help calm the coughing. Both the surgery and the follow up treatment are very harsh on a throat, and the chronic cough would be very irritating for the throat too! Soothing the throat during the day with liquids and lozengers, and maybe sleeping with a humidifier at night, might help with the coughing and aid the healing.
You might also want to consider having a videoflouroscopy to make sure the coughing isn’t related to swallowing issue. Certain consistencies might be causing residue that could be causing your mother to cough. Some people have issues with coughing after drinking the high calorie Ensure/Boost drinks because of the residue left after the swallow.

You might also want to notice if the coughing at night is related to your mother swallowing water when she brushes her teeth right before she goes to bed. If there is a delay in swallowing abilities, this could result in coughing when she lays down. (Your mother should wait at least 30 minutes before laying down, after eating or drinking.) Swallowing concerns are not always considered, however it is an easy thing to rule out as a possible cause for excessive coughing.

You might also want to check your mothers medications for side effects of coughing or swallowing concerns; many medications have dysphagia and dry throats as a side effect; the surgery might have changed her ability to manage it and it is causing the cough.
Good Luck!

I hear your concern. This must all be so unsettling for you. Thyroid surgery affected my vocal cords as well, including some mild paralysis of a few nerves. This resulted in hoarseness and coughing for me as well.
I am assuming your mother's drs are following up on her cancer levels with blood tests etc. I would think It's a little early for a secondary one to be affecting this, but understand why that would be a concern of yours. That's normal when going thru the difficulties of cancer.
I would suggest discussing Benzonatate, Tessalon Perels with your mothers doctors. If she is able to try them, they could help. They reduce the urge to cough , but if she has swallowing issue, may not be reccommended
Additionally trying to elevate the head of her bed, so she sleeps at an angle to reduce any reflux that may be contributing to it. Or sleeping in a recliner?
Hoping it eases soon!

I put a pillow under the top of my mattress and sleep with a humidifier. I also had a cough suppressant prescribed. This was for a bad cold/cough that eventually got a little better but is still lingering. I did have thyroid surgery 30 years ago and was left with 1/2 a thyroid and a diagnosis of a minute amount of papillary cancer. I used to get annual ultra scans to check for nodule growth. It's been 30 years and now I have this cough, but it has gotten better. Hope your mother gets better soon. Debbie