I think I submitted versions of a couple of these in the ER cookbook, too, but they are easy to look up on the 'net:
Linguini with Tuna Sauce
Zuppa di Farro
Minestrone is a classic: beans, broth, bit o' tomato product, then throw in whatever other veg. you like. They even have frozen packets of the basic unseasoned bean/veg. combo sans tomato only for this purpose in the supermarkets and sometimes you can find it on sale.
There's a pretty yummy Sicilian-style pasta dish w/sauteed cauliflower, saffron, pinoli, & raisins.
Bread --Rome has generally ASTOUNDING bread; more so if you go to the small alimentari or bakeries that supply them-- with a variety of cheeses, roasted veg., chopped tomatoes, garlic.. etc. makes a good late night supper / hanging-out spread. Ask for pane di semola (semolina) and it will keep fresh longer. Pane di Lariano is amazing.. partly a kind of whole-wheat but NOTHING like what we think of whole-wheat bread. We try to "import" this from Rome every chance we get. $plurge with some "affettati" (cold cuts). You can fry/grill some eggplants or roast some peperoni and covered with oil (bit of garlic if you like) they'll keep in the fridge for a loooong time.
Omelets are cheap. Add any/all of the ingredients above. Go for the no-name medium eggs if you can find them; in Italy, they cost 1/2 as much as the brand-name large.
There are any number of "salads" (cold or room temp. dishes) you can make with chickpeas (ceci), rice, or grains like farro. Add some drained canned tuna and some olives/chopped veg. For regular salads, the pre-washed bagged is expensive. The heads you have to wash are super-cheap (but need a lot of soaking & rinsing).
Send them to Piazza Vittorio / Esquilino. This is near a metro station (Vittorio), and kinda near the main train station (Termini). It's the Chinese/ethnic area and they've got a BIG street market zone. Lots of cheap produce and, in the little stores around there, you can, for example, get Chinese-type ("regular") long-grain white rice sold in bulk for .50/kilo instead of paying €2/kilo and up for the Italian arborio, which is all you'll find in the regular supermarkets. They might get sick of Italian food after a while.. so a stirfry could hit the spot for a change. The produce is more seasonal than in the US, so ask around. The price for something like tomatoes or fennel in season is like 1/4 the price the rest of the year.
If they develop a taste for fresh anchovies and sardines, those are very inexpensive as seafood goes. Pasta con le sarde is a great (Sicilian/southern) dish.
If they get to make friends with any of the locals, I'm sure they can get steered in the right direction.
A big budget-buster can be going out. For those who like to imbibe, cocktails can be shockingly expensive. At an outdoor music festival in Rome last summer I got charged €6 for a mojito ($8.50!!). Better save your pennies for the very reasonable (too reasonable!) jug wines and drink at home w/friends: can't miss w/a Sangiovese. Of course they won't be doing any of that, they'll be drinking acqua minerale -excuse me- acqua dal rubinetto (from the faucet) and studying, right??
A good cheap eats outing could be a bus trip to any of the "Castelli Romani" towns, or maybe Ariccia. GREAT roasted pork (porchetta), excellent bread, and cheap local white wine (Frascati). Need I say? Avoid places that look too flashy/touristy and check out the hole-in-the-wall joints. If they are in Rome, at some point a train trip to Naples could yield a whole 'nother gustatory (and cultural) experience.
Food in general is not usually cheap, in particular as it is not usual to find US-type sales/promotions/discounts. Meats are nowhere near as cheap as in the States, but, as ever, hotdogs are always a fallback. In Italy I've even seen hotdogs on pizza and sliced into whitebread sandwiches loaded w/mayonnaise (barf). A more classic homestyle meat preparation is "whatever" roasted over potato chunks with abundant salt, rosemary + garlic. Maybe black pepper, but that is suspect.
When I was studying Italian language, I found a very 'simpatico' bar/lunch spot just around the corner from Piazza dell'Orologio (facing the piazza v. North (ish), just a few steps away on the street to your back right-- could be via del Governo Vecchio-- no matter, it's the only "bar"/lunch spot nearby, where they made POP (pay -kinda?- one price.. it was never clear) sandwiches with EXCELLENT pizza romana (a kind of raised pizza dough/foccaccia found ONLY in Rome and which fills my dreams) + your choice of fillings-grilled eggplant, mozzarella, zucchini, ham, tomato.. there were a dozen to mix + match and they'd pop it in a sandwich grill to add the final crusty touch. They are/were very friendly to foreign students. Hope they are still there; the city center is unfortunately emptying out of "regular folks" and what remains are no longer apartments regular people live in and shops regular people shop in or restaurants regular people eat in. Mostly investment/foreign tourism anymore.
Italian cuisine is fairly parocchial, so they may have to make the best of what they can find. The good part is, it is usually of high quality in season, and the Italians tend to go for simple preparations where the ingredients stand on their own. The bad part is, the choice will not be as vast as in the US. Your boys will have a hard time living on Ramen noodles ($1.50+/packet in the regular supermarkets near me) and peanut butter (if they can find it and it is edible)... Popcorn is around, and a good cheap filling snack: a handful in a covered pan on the stove with a spoonful of oil, a grating of parmigiano while it's still hot, yum.
For pizza.. lascia perdere!! Per favore! While it may not be as cheap as eating in.. you won't get better pizza anywhere else other than Naples. They do have frozen pizzas (yes, bearable in a pinch) but unless they want to follow the true committed pizza-makers' path.. it's better to save the pizzas for eating out (not least because any other kind of restaurant --other than perhaps a wan, sorta-Cantonese-- is $$$).
Sorry.. Rachel Ray doesn't quite do the trick 4 me.. Any Italian would have an apoplexy over a parsley pesto ;-). If you served it them, you'd be their enemy for life.
The cheapest great pasta dish in assoluto? Spaghetti "aglio olio peperoncino". Cook spaghetti al dente in SALTED water. Sorry, salt is key. In the meantime in a big frypan (padella), with a decent amount of olive oil, let's say for 4 portions at LEAST 6 tablespoons.. Glug glug glug glug glug from the oil bottle.. Saute' at least a couple cloves chopped garlic (get a garlic press- indispensible) for a minute or so, maybe not even if the fire is hoppin', then add a bunch of hot pepper (peperoncino/cayenne) powder or flakes and heat enough to get the pepper woken up. Don't let the garlic get brown -max. lightly golden- then take the whole padella thing off the heat. When the pasta's ready, add to the padella and stir. If there's not enough oil.. add more! The true, late-night spaghettata.