Thyroid cancer (TC) is a very common malignancy worldwide. Chief among the innovative molecular drug targets for TC are epigenetic modifications. Increased telomerase activity in cancer cells makes telomerase a novel target for epigenetic anticancer drug innovation. Recently, telomerase reverse transcriptase (TERT) gene promoter (TERTp) mutations (C228T and C250T) were reported at high frequency in TC cell lines and tumor biopsies. In this study, three representative TC cell lines, mutant TERTp (TPC1), mutant BRAF/TERTp (KTC2), and wild-type TERTp (WRO), were screened with a drug library composed of 51 epigenetic drugs: 14 Aurora kinase inhibitors; 23 histone deacetylase inhibitors; 5 sirtuin modifiers; 3 hypoxia-inducible factor inhibitors; 2 DNA methyltransferase inhibitors; 2 histone methyltransferase inhibitors, a histone demethylase inhibitor, and a bromodomain inhibitor. Effects of the drugs on cell growth at 48 and 72 h were compared. PF-03814735, a small-molecule inhibitor of Aurora kinase A (IC50 = 0.8 nM) and B (IC50 = 5 nM), was the most potent on KTC2 cells, whereas CUDC-101, a multitarget inhibitor, was effective on both WRO and KTC2 cells. Notably, PF-03814735 was found to be the most effective epigenetic drug on cell lines harboring the C228T mutation. In conclusion, these new findings offer specific guidance on dose and time course selection to design novel therapeutic interventions against TC using PF-03814735, and specifically target cells carrying the TERTp(C228T) mutation. In a larger context of drug discovery science, these findings inform new strategies to forecast optimal treatment regimens for TC, particularly with Aurora kinase inhibitors and in ways guided by epigenetic drug design.