“I Won’t Be a Minute” Means “I’ll Be Quick”

In everyday Australian speech, I won't be a minute is a natural way to say that you are stepping away briefly or asking someone to wait. It means “I'll be quick” or “I won't take long”. A minute is not a promise of exactly sixty seconds here; it means a short period and gives the listener some reassurance.

At the office, you might say: “I just need to grab my laptop. I won't be a minute.” The other person expects you back soon, but is unlikely to start a timer. Do not confuse it with I don't have a minute, which means you have no time available now. If you expect a long delay, I won't be a minute is not the right reassurance.

https://dictionary.cambridge.org/dictionary/english/minute