Unclear.
This is usually considered to be a loan from 油 (MC yuw). There are two points supporting this stance: one, the word was spelled with 油 in Nôm texts and dictionaries (although this point is not absolute due to the word's unfortunate absence in important texts such as Phật thuyết đại báo phụ mẫu ân trọng kinh (佛說大報父母恩重經) and Quốc âm thi tập (國音詩集)), two, in Dictionarium Annamiticum Lusitanum et Latinum (1651), a dictionary based chiefly on the Northern dialects and an important document on the then-disappearing contrast between ‹dĕ-› (< Proto-Vietic *-t/d-) and ‹d-› (< PV *j-) in these dialects, only dầu is attested, while *dĕầu is not.
On the other hand, as Jacques (2022)[1] argued, there are indeed forms such as Kha Phong ntuː¹ and Thavung atuː¹, for which Ferlus (2007) reconstructed Proto-Vietic *-tuː (“oil”). Furthermore, there was a word spelled ‹taw›, with the English equivalent ‹Oil› and Chinese equivalent yeo, attested in Barrow (1806)'s A voyage to Cochinchina, in the years 1792 and 1793, recorded either a very aberrant Vietnamese dialect or a Vietnamese-based pidgin of Turon (modern day Đà Nẵng). The name of thầu dầu (“castor oil plant”) (from etymology 2 below) is also likely related, and it has the Tho cognate tʌw² tʌw².
Likely the same as etymology 1. Dipterocarpus spp. are oil producing plants.
(classifier cây) dầu