Location is by far the most important thing for the budget traveler. Not only does the proximity of your hostel to city center/attractions usually determine the cost per night, but the choice of which cities to visit (and how many) can be a major determinant of the cost. It's already been pointed out that Galway is cheaper than Dublin, and anywhere is cheaper than London, but you should also consider how many places you want to visit on the trip. While it's great to see a lot of different places, each new location trades off with time getting to know one city even better, and each trip costs that much more money.

This isn't to say that you can't get around Europe for cheap. I've seen ride-shares advertised on couchsurfing, and I hear craigslist is active in some European cities as well, but pay-for travel always has added costs. Reservations on top of rail passes (though you can often avoid this if you're willing to risk not getting onto a train - check the details for your route) and the taxes, bag-check fees, and out-of-the-way airport costs make the budget airlines less of a bargain than the .01c sticker price appears (though still worth it!).

There are advantages to the beaten path - the sites are likely to be open, you'll have little trouble getting around if monolingual, hostels will rarely be in short supply, you'll always be with a community of travelers - but you trade off so much. When traveling the standard Western European routes, or the recent Eastward expansions, in the peak summer season, you'll be paying more, dealing with bigger crowds, and seeing less of the place your visiting and more of the tourist culture that's taking over it for the time. Not that this makes Cinque Terre in July a bad decision, it just means that you might want to consider the Dalmatian coast of Croatia instead, or even Montenegro - a country functioning on the Euro but affordable nonetheless.