When someone says "a minute", they usually don't mean exactly 60 seconds. They mean a short length of time, somewhere between 30 seconds and 5-10 minutes.
If you want to specify exactly one minute, use the phrase "one minute". The sentence:
Can you give me just one minute to finish writing this?
...sounds much more specific about how long you are going to spend writing the message.
It's common to add "just" to this phrase:
A: Are you coming?
B: Yeah, just a minute!