From my understanding Icelandic opinion is roughly a 50/50 split on whether they should join the EU or not, and recent events such as Trump threatening to invade Greenland are likely to influence the outcome of a referendum.

The current administration has the long-term goal for Iceland to join the EU, but is willing to wait it out till a good opportunity shows itself before it pushes for EU membership. Them pulling the referendum forward, would suggest that they believe public opinion right now is in favour of EU membership, so this is the best chance they have of the referendum going their way.

A big sticking point for Iceland is the common fisheries policy of the EU. Fishing is a significant part of both the Icelandic economy as well as their culture, and joining the common fisheries policy will probably be highly disruptive for them.
If the EU is willing to give Iceland an opt-out on that, that would certainly speed things up.

Source: My partner is from Iceland, and we occasionally discuss the politics there.

Iceland is already more integrated in the Union than the UK ever was. They should have a say on the discussions that very directly influence them. Same for Norway btw.

That's called being a member of the EU.

Honestly, wouldn't mind giving them that opt-out. They also should be able to negotiate favourable LGBT+ and labour rights protections, in case.

I wouldn't say they are more integrated than the UK was. Iceland is not a part of the customs union, and I would argue that that is a pretty important part of the whole concept of the EU.

They are also not part of the common fisheries policy, which is going to be a pretty big hurdle to cross.

No chance the EU agrees to that.

True, but they're part of Schengen, symbolically and also from a human perspective, a much more powerful policy than fisheries or customs..

Note: this is a referendum on whether to restart the negotiations. I assume a second one would be needed if this one passes and negotiations are concluded.