Iceland sounds perfect for you if your interests are mountains, nature, and fun activities to do in the summer time. The vast majority of Iceland is mountainous, the nature here is unbelievable (I moved here after I came for the first time, just fell in love with it), there's tons of fun activities (incredible party, music, and arts scenes, for example), and as for summer time, that's when all the massive festivals are and it never gets dark.
Flights too expensive though? From the UK? Boggle. I just went to Icelandair's website (Icelandair being usually more expensive) and picked early July, and tickets are around £240.42 per passenger including tax. Is that too expensive for you(???). I then tried June and found tickets for £186.42 per passenger - even cheaper.
Update: Summer weather depends where you are. If you're on top of a glacier, it's cold year round! But in the lowlands, generally, July highs are around 15-16C or so, give or take. It's a little warmer in the north than in the south during summer. Oh, and Iceland can be windy.
But seriously, £40 a ticket? Where can you fly for £40 a ticket? How can an airline keep an airplane in the air for £40 a ticket? How can you afford food, lodgings, local travel, activities, etc anywhere if you can't afford more than £40 a ticket?
For example, someone mentioned Bucharest. I went to cheaptickets and looked up tickets from London to Bucharest in June, and gave it as flexible options as possible. It said the tickets are $280 USD, which is £170 (compared to the £186.42 I found for Iceland in June).
The problem isn't that tickets to Iceland are expensive; it's that your expectations are unreasonable. I mean, seriously, £40 a ticket? That's, what, 6 hours work at minimum wage? And you want that to cover 7 hours (round trip) of air travel time?
If you really have that little money, you should probably forget about international travel and hop on a train out to the British countryside for your vacation. From my experience with the UK, the landscape (and the people!) get nicer the further away from London you get.
Update: I went and used your site. The cheapest I found for a June trip (I picked the 6th to the 20th) was £80 to Bucharest. It showed £103 to Iceland on RyanAir. Not exactly a huge difference, percentage-wise. And I'm not sure whether those prices include *all* taxes/fees, and they certainly don't include the regular airline stuff like baggage fees. And again, I reiterate, if all you can afford is £40 a ticket, how do you expect to *live* there and handle domestic travel once you get to your destination, if you have so little money? Tickets are a one-time cost. You have to pay for food, lodgings, etc every day, and often domestic travel as well. Are you planning to wildcamp and eat from a dumpster? If your ticket budget is £40 then surely your daily budget can't be more than £20. That wouldn't even pay for a bunk in a crowded hostel in most parts of Europe, much less *all* of your daily expenses.
@NiNL: First off, Iceland isn't a two hour flight, it's 3,5h. Secondly, you're buying *two* tickets when you fly round trip, so you have to double the flight length (7h, not 2h). Third, your own personal choice of destinations, Slovenia, using the asker's own choice of discount flight-finding websites, skyscanner, for the above timeperiod, shows £112 per ticket (same caveats as above), even more expensive than to Iceland. Those "£5 tickets" you hear about, first off, they're only last-minute surplus tickets (see below), and secondly, that's not including the (significant) taxes and fees that they add on top of it when you try to buy.
I once again maintain that the problem is that the askers' expectations for ticket costs are absurd for international travel, not their choice of destination.
@Dariana2: Picking dates in January will inherently distort your figures, as you can sometimes get slots on underbooked flights that will be leaving shortly that the airlines want to fill up. You can't do this when planning months ahead. If the asker wants to base their trip on a "maybe", then by all means, they can wait until the last minute and hope there will be a discount ticket. Or the only ticket available could be £600. It's the luck of the draw.
I have no clue what your "random date in June" is nor what site you used, so I can't either double check your figures nor get a comparison Iceland ticket for that timeperiod.