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.