I went in March 2016. When I was there, I took a tour (I think it was with Extreme Iceland) which included the Golden Circle, a blue ice cave and glacier hiking.
I’m not sure that I picked the best time to go, although I suppose it’s always a trade off:
The sights in the Golden Circle were quite crowded, even in March.
We saw Seljalandsfoss (the waterfall you can walk behind), but the path behind the curtain of water was closed, due to the spring thaw making it slippery.
We were one of the last groups that winter who were able to visit the blue ice cave we saw. I was very glad not to miss it, but I’m told it was a much more impressive cave earlier in the season.
We didn’t really see the NL, just a colourless wisp in the sky that showed up green on a long exposure photo.
On the plus side, in March we had enough hours of daylight to see plenty of sights and the sky was dark enough at night to potentially see the NL. The main sights must have been less crowded than in the summer and although the weather was cold (and wet at times), it was easily bearable with the right clothing.