As John Cavan notes, it's common for elderly cats but we don't currently know what the cause of hyperthyroidism is. I'll outline some of the current research directions here.
The research is not yet conclusive for anything (and at times, conflicting!), but these are the paths that medical science is investigating.
This study is a broad survey (263 cats) to try to determine differences between hyperthyroid and normal cats.
J Am Vet Med Assoc. 2000 Sep 15;217(6):853-6. Evaluation of dietary and environmental risk factors for hyperthyroidism in cats. Martin KM1, Rossing MA, Ryland LM, DiGiacomo RF, Freitag WA. web abstract
Case cats were significantly less likely to have been born recently
than control cats. Housing; exposure to fertilizers, herbicides, or
plant pesticides; regular use of flea products; and presence of a
smoker in the home were not significantly associated with an increased
risk of disease, but cats that preferred fish or liver and giblets
flavors of canned cat food had an increased risk.
Diagnoses of hyperthyroidism starting becoming a lot more common in the 1970s.
J Feline Med Surg. 2010 Sep;12(9):672-9. doi: 10.1016/j.jfms.2010.07.011. Feline hyperthyroidism: potential relationship with iodine supplement requirements of commercial cat foods. Edinboro CH1, Scott-Moncrieff JC, Glickman LT. web abstract
A review of historical iodine recommendations revealed that the units
of iodine supplementation changed in the 1970s. Given this change,
foods minimally supplemented since the late 1970s would have been
iodine deficient for most cats.
PRACTICAL RELEVANCE: Iodine supplementation of commercial cat foods
should be evaluated in the light of the iodine recommendations revised
in 2006. Foods may remain deficient in iodine if supplemented at the
minimum recommended concentration, possibly contributing to the
development of FH.
This study looked at iodine concentration of different commercial food.
J Feline Med Surg. 2013 Aug;15(8):717-24. doi: 10.1177/1098612X13477855. Epub 2013 Feb 25. Iodine concentration in commercial cat foods from three regions of the USA, 2008-2009. Edinboro CH1, Pearce EN, Pino S, Braverman LE. web abstract
Dramatic variation among canned foods (resulting in ingestion of approximately 49-9639 μg iodine/day) suggests that the disparity in iodine concentrations may lead to development of nodular hyperplasia and, later, clinical hyperthyroidism, if cats consume diets that are at first iodine-deficient and later contain excessive iodine. Manufacturers are encouraged to ensure adequate iodine supplementation across all products and areas of the USA.
Flame Retardants (PBDEs)
Arch Environ Contam Toxicol. 2012 Jul;63(1):161-8. doi: 10.1007/s00244-012-9750-y. Epub 2012 Feb 5. Decabromobiphenyl, polybrominated diphenyl ethers, and brominated phenolic compounds in serum of cats diagnosed with the endocrine disease feline hyperthyroidism. Norrgran J1, Jones B, Lindquist NG, Bergman A. web abstract
It is notable that BB-209, 6-OH-BDE47, and 2,4,6-tribromophenol all suggested that endocrine-disrupting chemicals were present in high concentrations in cat serum
J Toxicol Environ Health A. 2012;75(4):201-12. doi: 10.1080/15287394.2012.652054. The feline thyroid gland: a model for endocrine disruption by polybrominated diphenyl ethers (PBDEs)? Mensching DA1, Slater M, Scott JW, Ferguson DC, Beasley VR. web abstract
In dust from homes of hyperthyroid cats, total PBDE concentrations were significantly higher, ranging from 1100 to 95,000 ng/g. Dust PBDE and serum total T4 concentration were also significantly correlated.
Environ Toxicol Chem. 2012 Feb;31(2):301-6. doi: 10.1002/etc.1700. Epub 2011 Dec 29. High polybrominated diphenyl ether levels in California house cats: house dust a primary source? Guo W1, Park JS, Wang Y, Gardner S, Baek C, Petreas M, Hooper K. web abstract
The authors found no evidence that linked levels of PBDEs, PCBs, or OCPs with hyperthyroidism. This may be because of the small sample size, competing or confounding risk factors, or complicated causal mechanisms.
Jessica Norrgran, Bernt Jones, Anders Bignert, Ioannis Athanassiadis, and Åke Bergman
Higher PBDE Serum Concentrations May Be Associated with Feline Hyperthyroidism in Swedish Cats
Environ. Sci. Technol., 2015, 49 (8), pp 5107–5114, DOI: 10.1021/acs.est.5b00234, Publication Date (Web): March 25, 2015
hyperthyroid compared to euthyroid cats have higher serum concentrations for some of the investigated PBDEs (BDE-99, BDE-153, and BDE-183) and CB-153 on a fat weight basis.
Honestly I can't really make heads or tails of this one, but I'm including it for completeness
J Vet Intern Med. 2011 Sep-Oct;25(5):1057-65. doi: 10.1111/j.1939-1676.2011.00790.x. Epub 2011 Aug 30. Evaluation of predictors for the diagnosis of hyperthyroidism in cats. Wakeling J1, Elliott J, Syme H. web abstract
This abstract doesn't say much, but I suspect the contents of the article would be very interesting:
J Feline Med Surg. 2012 Nov;14(11):804-18. doi: 10.1177/1098612X12464462. Hyperthyroidism in cats: what's causing this epidemic of thyroid disease and can we prevent it? Peterson M. web abstract