What is the cheapest form of public transportation from Amsterdam to Northern Ireland?

On megabus.co.uk, you can find an Amsterdam - London ticket for 17.50 pounds. Then, a London to Dublin ticket can set you back as little as 15 pounds. Both connections include a ferry.

Occasionally, Megabus offers deals of 10 pounds for each of the above legs. But, note that trying to book the whole journey as one trip is almost always (if not always) more expensive.

Dublin to Belfast can be had for 8 euros via Bus Eireann.

This is certainly not the most pleasant method of travel, but it's damn cheap. If you're lucky, and flexible with your dates, you might get a Ryanair flight from Eindhoven to Dublin for as little as 18 euros. But then you still need to do some peripheral travel.

Your cheapest option is probably a flight, but this is not certain. Easyjet appears to have cheap flights between Amsterdam and Belfast, although I'm not sure how many hidden fees are added to their advertised price. If not flying, between Amsterdam and London, a cheap option is the Dutch Flyer, which will set you back £49: Dutch Flyer connects you from any train station in The Netherlands to any in East Anglia including/as well as London Liverpool Street station. Flying may be cheaper but not usually when counting all additional fees and transportation to and from the airport.

Between England and Ireland, Seat61 has a rather complete listing. Apparently, fares start as £39 from London to Dublin by train+ferry, or 53£ to Belfast. It appears Busbud has tickets from 42£.

You might find cheaper options using buses, as per the answer to the Amsterdam-Paris question, although it seems Eurolines does not serve Belfast, National Express does (but not Amsterdam). So by bus you'd need at least two different tickets, which makes finding the cheapest option considerably more difficult.

I just checked several flight options.
EasyJet, if you do not need to bring luggage, starts from €32, a few months from now. That is including all taxes but excluding all extras you can avoid, like luggage and from Schiphol, so €2.10 for a bus ticket needs to be added. Not sure how much extra for the Belfast end, Belfast International.

Eurolines London to Belfast on the day I tried is already £30, which is quite a bit more than the basic price for EasyJet Amsterdam to Belfast. And you need to get to London first.
Megabus does not seem to include Belfast these days. But when booking on short notice the price do not get much higher while flights do get more expensive at the drop of a hat.

In short, search all options, the final price depends on the day you book more than on the actual mode of travel.