Does your tsh (thyroid) level change during the day? how much?

Yes: Tsh peaks around midnight and is higher in the early morning than in the afternoon. In a normal person these changes are quite small and were not even detected until the 3rd generation TSH test became available.

No clinical issue: While there can be some mild fluctuation in TSH detected by sensitive assay, this is of no clinical significance to the patient over the course of a day.