without breaking the bank - still in college
#1 is right about what is amazing in Iceland, but it is difficult to take advantage of these things without your own transport, especially in the off-season - Summer is very short in Iceland. You can also do thrill day-trips out of Reykjavik, but those are mostly excruciatingly expensive. You will find Iceland in general excruciatingly expensive, especially coming from the W side of the Atlantic. It is also difficult to travel in without your own transport, because transport is thin on the ground and very expensive, including buses which are expensive, and take you to settlements not tourist attractions, unless they happen to be on the way. Transport and tours is the main killer budget item for intended-low-budget travellers. And the range of transport options is small in May. Hostel accommodation is not unreasonably expensive by European standards in Iceland these days, and food in supermarkets likewise - Iceland isn't quite as bad as Norway or Switzerland for those these days, but the latter at least have rather more in the way of transport. Whilst camping is the cheapest option, many campgrounds are not open yet, and some years winter isn't really over yet in early May, so camping could be a cold and lonely experience at that time of year.
You may be better off spending a bit more on the plane ride and going somewhere cheaper and where there are more services for low-budget travellers, and where winter is properly over. May is lovely in Mediterranean Europe for example. I realise your family probably like the idea of you going to a super-secure country like Iceland, but travelling in most of Europe isn't going to be much different from, and in many cases better than, travelling in your own country from a security aspect.