Possibly, but only Niantic knows when. I would strongly discourage waiting for it.
The thing about usernames and gaming is that you need more than confirmation that the original account has been closed. You need the developers to release the actual username.
Until Niantic actually release the username, their servers will still register it as taken, even though it points to a 'blank' user. Companies have been known to release usernames, in the past. That said, it is often the result of years and years of activity, where the game has run for so long that there would be a large number of 'deleted' accounts, and usernames that could be recycled.
There are two options available to you, in terms of obtaining the user name again, in the short term.
If you contact the developers directly, there is no guarantee, but they might very well give you the ability to 'pass on' your username. Be cordial, and ask them nicely, as opposed to demanding they do it. Enforce that the username currently belongs to your account, that you would like to give it to your friend, and that the new username would make their experience more enjoyable.
Ultimately, developers are always more capable of acting if they think it would make a bigger difference to their users, but much more goes into play. Do not be surprised if they reply to say 'Sorry, but such action is outside of our capability' or 'Sorry, but that would not be fair to other users'.
Give your friend your account
This section had previously detailed options for giving your friend your account. That said, Niantic has since updated their FAQ to enforce that this practice is considered cheating by their standards, and will result in account termination, so it probably is not the best course of action.