Thyme is an evergreen herb with culinary, medicinal, and ornamental uses.
The most common variety is Thymus vulgaris.
Thyme is of the genus Thymus of the mint family, and a relative of the oregano genus Origanum.
Thyme is sold both fresh and dried. While summer-seasonal, fresh greenhouse thyme is often available year round.
The fresh form is more flavourful, but also less convenient; storage life is rarely more than a week.
Although the fresh form only lasts a week or two under refrigeration, it can last many months if carefully frozen. (source)
So can hamsters eat thyme?
Yes it is fine for them to eat.