Whilst means during the time that, in formal British use. Whilst waiting, whilst it lasted, whilst I agree.
It is interchangeable with while and is now almost exclusively British — American writing uses whilst about never, and some British style guides advise against it as fussy. It carries a formal register, which is why it turns up in letters, notices and academic writing more than in speech. Note amongst and amidst, which are the same pattern; and that all three -st forms are historical additions rather than the older word.
1prepositiontimeDuring the time that.
“Someone stole my bag whilst I was asleep on the train.”
2prepositionwhilstAlthough, in formal use.
“Whilst I understand your concerns, I cannot agree to change the contract.”
Too few real examples of this word to estimate which sense is commonest.
whilst is a conjunction of time.
Whilst is whiles plus an added -t, and whiles was itself a genitive of while. The same excrescent -t produced amongst, amidst and against, none of which needed it.
Proto-Indo-European
*kʷyeh₁- — to rest; peace, rest
Proto-Germanic
*hwīlō — period of time, a while; period of rest, break, pause
Old English
hwīl — period of time, a while
Middle English
whilst
English, today
whilst
Built from
From Late Middle English whilst, whilest, qwhilste Northern England, quilest Northwest Midlandswhylest, whylst, (Northwest Midlands) quilist, quylest, quylist, from whiles (“during the time that, while; only so long as; provided that; because, since; until”) + -t. Whiles is derived from whiles (“period of time, a while”) (probably from the second element of adverbs and conjunctions like otherwhiles and somewhiles), from while (“period of time, a while”) + -s; and while is from Old English hwīl (“period of time, a while”), ultimately from Proto-Germanic hwīlō (“period of time, a while; period of rest, break, pause”), from Proto-Indo-European kʷyeh₁- (“to rest; peace, rest”). The English word can be analysed as whiles + t. 2
Etymology from Wiktionary, CC BY-SA