The second-to-last post here claims that
Actually, the plant used in the movies is white thyme. It grows everywhere, can be used medicinally, is considered an invasive plant and is also given to the pigs to eat. It is also found in purple
However, the images they show are of Thymus serpyllum (Wikipedia), whereas the only references I can find to invasive thyme in NZ are about Thymus vulgarum (wild thyme) (Wikipedia), e.g. here.
The last post says it's common myrtle (Myrtus communis).
Hard to tell scale from the screenshot: this picture suggests the flowers are quite small (~ 1 cm in diameter?)
- top left: LoTR still capture (from https://lotr.fandom.com/wiki/Athelas?file=Athelas.jpg)
- top right: common myrtle, Giancarlo Dessì, CC-BY-SA, https://commons.wikimedia.org/wiki/File:Myrtus_communis10.jpg
- bottom left: elfin thyme, KENPEI, CC-BY-SA, https://en.wikipedia.org/wiki/Thymus_serpyllum#/media/File:Thymus_serpyllum_var_albus1.jpg
- bottom right: wild thyme, Ferran Turmo Gort, CC-BY-NC-SA, https://inaturalist.nz/photos/48088
New guess: https://identify.plantnet.org/ thinks Bacopa repens (Sw.) Wettst. Creeping waterhyssop. But is the habitat wet enough ... ??
Gerlinde(More guesses from PlantNet: Acleisanthes longiflora A.Gray (Angel trumpet), Silene pinetorum Boiss. & Heldr., Bacopa monnieri (L.) Wettst. (Monnier's hedge hyssop) ...
Water hyssop from Gerlinde (cc-by-sa) - Sep 13, 2021: