I agree with everything said so far but would add a few thoughts:
Doing preliminary lodging research on booking.com can tip the group off to destinations likely to be considerably more, or less, expensive than others. Some hostels are listed there, but certainly not all. The temptation on a first trip is to hit the highlights (I did that myself in 1972), but if they want to be frugal, they do need to resist the temptation to make this a trip focused solely on capital cities and other very famous destinations (like Venice, Florence and Barcelona). Maybe they can make stops in places like that rather short and include some interesting, less costly stops.
Two countries with endless attractive smaller-town alternatives that aren't crazy-expensive to visit are Germany and Italy. (They just need not to stop off in Switzerland along the way!) Italian trains are less expensive, per mile, than those in many other countries, and there are a lot of great places to visit in every region of the country, making use of cheap-but-slow regional trains not only practical but also necessary.
Germany has a stunningly good deal, the Deutschland Ticket, that covers essentially all local trains and buses (not including the express trains) nationwide. The cost will be 58 euros per calendar month for next year. With that ticket you can travel the entire country (just avoiding the express trains), including city transportation. There have been several threads on this forum about the best place to buy that ticket (you have to know how to cancel it, or you'll be charged for another month), but this website provides some useful descriptive information in English. I have no idea whether it's a good place to buy the ticket. https://www.rmv.de/s/en/deutschland-ticket
It would be smart to warn the travellers that the locals they see hopping on and off city buses and trams probably have monthly or annual tickets in their wallets. Not seeing those tickets being flashed doesn't mean the local transportation is free. Tourists need to do a bit of research so they know how to pay (in some places that means buying a ticket before boarding) and knowing how to validate it. I mention this because there are ticket inspectors roaming around, keeping passengers honest, and being caught without a valid ticket will generate a fine of (probably) at least 50 euros. No excuses accepted.
If they decide to take as long a trip as their schedules allow and know in their hearts they're going to want to move around a lot, hitting the highlights (doing something like London-Paris-Barcelona-Madrid-Rome-Florence-Venice-Vienna-Budapest-Munich-Berlin), that will be a lot of miles, and it's possible a Eurail Pass would be produce savings. The 2-month Youth Eurail Pass is on sale for $493 until December 17. The one-month pass is on sale for $416. The shorter passes are a lot more expensive per day and I think much less likely to pay off. They need to know that the express trains in many countries require seat reservations, and those are an extra cost. If you buy point-to-point tickets on such trains, the seat reservations are included in the basic ticket price. This makes it tricky to compare the cost of a rail pass (to which seat-reservation fees must be added) to buying point-to-point tickets.
I'd keep the existence of budget airlines in mind. If they decide to visit non-contiguous countries (England-France-Spain-Greece, for example), it could make a great deal of sense to fly one leg of the itinerary. A Eurail Pass becomes much less attractive in that situation.
Buying multi-city rather than round-trip transatlantic tickets will probably save them at least one full travel day, and it may not be much more expensive than round-trip tickets. It depends on the specific cities selected.
I recommend the immediate purchase and reading of Rick's "Europe Through the Back Door".