Probably a lot more than £1k. We spent that on accommodation and flights alone for two of us when we went for 4 days a couple of years ago and that was mid range pricing - could have spent a lot more but it would have been hard to get it much cheaper.
Then there was food and excursions on top, which probably at least doubled the cost.
But to answer the OPs question, yes it definitely is worth it and you can pack a lot in 4 days.
We arrived early on day 1 and spent the day wandering around Reykjavik, on day 2 we did the Golden Circle tour, day 3 visited a few museums in Reykjavik and went swimming in one of the many geothermally heated outdoor pools, which had slides and hot tubs, day 4 we went whale watching and then we flew home.
Would definitely go again, at different times of year too - we went in September, but would also like to go in winter for the northern lights and long dark nights and also like to go in mid summer. And spend some time travelling around more.
But at 17, unless it's something you can easily afford, I'd expect him to contribute around half the cost, either from any money he has saved, in lieu of Christmas/birthday presents or by him working.