A fair few times in the past I have contemplated travelling to Norway, only to back out at the last minute when I realised how much it would cost.

I'm pretty close to booking up to head over there for a short trip this winter, but does anyone have any ideas on how to save a little bit of money

Any ideas on saving on train fares?
The Day trip from Bergen to Flam has pretty much been written off as it would cost €260 for 2 even with Student discount and advance purchase

London is suddenly looking cheap!

Cycle, wild camping and take your own fishing rod?

Not far wrong! We have travelled Norway twice over the years
Our "budget" approach was to use a Rentacar.
You can hire a small one with CDW/Unlimited Mileage for a week for c £250 +/-. It then can double as a hotel - all seats recline nowadays and a sleeping bag will fit in the budget airline's baggage allowance - or else pay for one suitcase. Take along a "boiler", plates/spoons etc and you are virtually self sufficient! Camping wherever you want in Norway within reason is quite normal
Whether Mrs Meltwaterfalls would be as willing to "wild camp" in a car as much as Mrs Solivagant was some years ago only you know!!
Buffet lunch is a "good deal" meal to buy at hotels/restaurants and breakfast materials can be bought from the supermarkets.
The odd youth hostel can be used if you need a "clean up"!
The big advantage for us beyond cost issues was that no time needed to be spent looking for hotels and wasting part of the day because one had to check in early - so we got up to North Cape and back in a week from Bergen. And of course you are totally free of public transport schedules.
Perhaps winter isn't the best time to do this -though we parked among the snowdrifts near Hammerfest even in late May!

Thanks for this, we looked into it a lot over the weekend.

I think we may shelve our plans again and perhaps go for a summer trip to take advantage of the ideas proposed.

One saving I did find was that there is no maximum distance for the MiniPris train tickets. It doesn't help with our proposed trip from Bergen to Flam (Flamsbahn is a separate entity so will cost additional), but you could conceivably take a day long train trip from Bergen/ Stavenger/ Oslo to Bodø in the Arctic circle for c.€30 which is pretty good value if you don't mind so long on a standard train. For us WHS hunters Røros is also available on this ticket.

Not sure how serious and efficient you want to see the country. I have limited holidays, so I specialized on effective travelling.

I only went there for a long weekend to see Oslo and Bergen and a Fjord on the way that belongs to the WH list. It cost me a lot. Car rental was reasonably prized. Petrol, hotels and even the food in super markets were incredibly high.

I guess an effective way would be to buy a fly pass from Wideroe. Grants you 2 weeks unlimited flying within the country for 2 weeks. They fly to Hammerfest (close to Alta), Roros, Sandnessjoen (close to the Vega Archipelago, Bergen, Bodo (close to the Lapponean Area FIN WHS) and some places near the big Fjords. For the Urnes Stave Church I'd recommend a car. I didn't have time to add the church on my return car trip from Oslo to Bergen, but I will do it by car in future.

I went in February and streets were in good shape. But I know from a friend that they cancel a lot of flights during winter due to snow storms and most important, days are simply too short and nights too long.

If you go for the train, keep in mind that you will spend a lot of time in trains to see all the WHS. Eg Oslo-Bergen over 7h, Oslo-Roros over 5h. The long travel times also mean that you will have to spend more nights in the country which means more expensive hotels and more expensive food to be paid.