The noun G. saith “hunger” appeared in the Gnomish Lexicon of the 1910s (GL/66), likely based on the early root ᴱ√SAẎA “hunger” which was mostly used in combination with -kǝ (QL/82). In Early Noldorin Word-lists the word appeared as sais, revised to ᴱN. hais (PE13/147, 153) to reflect the fact that initial *s* generally became *h* in the Noldorin of the 1920s (HIPTN/§4.1.1). In this 1920s document its Qenya equivalent was ᴱQ. saiste indicating a primitive form of ✱saistē.
Neo-Sindarin: For purposes of Neo-Sindarin, I use ᴺS. saeg for “hunger” based on the Neo-Root ᴺ√SAYAK, a neologism coined by Fiona Jallings based on ᴱQ. saike “hunger” (FJNS/352). An argument can also be made for ᴺS. saeth as a more direct update of G. saith (perhaps < sayak-tē as suggested by Elaran in 2018), but I prefer saeg since its etymology is more transparent.
A noun appearing as ᴱQ. saike (saiki-) “hunger” in the Qenya Lexicon under the early root ᴱ√SAẎA (QL/82). In Early Noldorin Word-lists of the 1920s, the Qenya word for “hunger” was instead saiste (PE13/147).
Neo-Quenya: For purposes of Neo-Quenya I would retain the noun ᴺQ. saicë “hunger” based on a Neo-Root ᴺ√SAYAK, but only as an archaic word so I could justify the retention of various famine-related words like ᴺQ. saicelë. For normal speech I would assume ᴺQ. maitië is the ordinary word for “hunger”, a noun form of the adjective Q. maita “hungry” from around 1960.