The giveaway option is limited to five days, which ought to be plenty. If they're really close, you could try three days.
You don't have to advertise that your book is free, though likely a few hundred people will discover it anyhow.
As an alternative, you can gift the book to them, which will cost you the amount of Amazon's cut, which is usually 30 percent (assuming you're in the 70 percent royalty program). If your book is $2.99, that would be 40 times 99 cents, less than the postage if you were to mail a copy.
Finally, you can send them a *.mobi file, which requires you to figure out how to create one, and requires them to figure out how to transfer it to their Kindle.