1. Use Travel Rewards Points to Get Flights to Europe.
Travel rewards are what got our family of five to Europe, and though some time is required to choose the right credit card and the right rewards system, this is time well spent if your family is able to wing its way overseas for a budget-friendly trip. As much as possible, no dollar gets spent in our household that doesn't accumulate a rewards point of one type of another; work-related travel also helps to accumulate miles.
About.com's Guide for Budget Travel can help you get started with finding the right travel rewards credit card: there are many factors to consider, including annual fees, expiration policies, whether the card is restricted to certain airlines, how well the available airlines fit with your travel wishes... Keep in mind, too, that some travel rewards systems require a user to pay certain taxes with their "free" tickets, and these taxes can be substantial depending on the destination. (London, for example, may be more expensive for airport taxes than an alternative destination that may work just as well for your trip.)
If you can't manage to get plane tickets with airpoints of some sort, check out these tips about finding cheap international airfares, at About.com's Budget Travel website.
2. Use Budget Airlines
Travel within Europe can be ridiculously cheap for travelers willing to fly into secondary airports (and with limited luggage.) Again, About.com's Guide for Budget Travel can get you started on using budget airlines within Europe.
3. Getting Around
About.com's Guide for Europe Travel has lots of advice about Getting Around Cheaply in Europe. One thing to consider is that -- if there are four or more of you -- renting your own vehicle may compare favorably with the costs of, say, four train tickets or Eurail passes. A great option if you're able to stay 17 days or more is the Renault Eurodrive "Car Lease Buy-Back": this is basically a car rental but with some excellent features, such as brand-new vehicles and zero-deductible insurance, as well as a good price point.
With your own vehicle, you're free to go where you want and also it's much easier to accommodate children's needs as they arise. You can also carry convenient gear around with you, and you can stockpile cereal, fruit, snacks, cheese, bread, and bottled drinks, which can save considerable money when feeding the gang.
4. Budget Accommodation
There are many ways to save money on lodgings in Europe, if you're willing to think outside the usual model of a North American- style hotel. Family-friendly hostels, for instance, can be an economical choice and also fun, sociable places to stay: click to see a charming picture of the Villa Saint Exupery Gardens in Nice, where families can book a private room with en suite bathroom, in a converted monastery in a quiet area. Often hostels offer character buildings, kitchen conveniences, board games and play areas-- read more about the good points of staying in a family-friendly hostel in Europe.
Other ways to save money on lodging in Europe include "camping villages" that offer bungalows with fully equipped kitchens, and large recreational properties with restaurants, activities, kids programs, and maybe even some extras such as a spa or waterslides. Families can also share a large "villa" vacation home with another family, or do a home exchange... Read about more ways to find budget-friendly lodging in Europe. Also check out Airbnb, a website where ordinary people offer rooms, B&B lodging or full apartments to rent; visiting families can enjoy the use of kitchens, multiple bedrooms, and maybe even cribs, high-chairs, etc.
5. Do Free Things
Finally, in many European cities families can have a wonderful time in without paying any admission charges. In many cities in Europe, one you've gotten the family there and found a place to stay, a huge amount of enjoyment and experience is there, for free. Simply walking along the historic streets is a treat, for starters. And there are are the free museums... In London, the Science Museum, Museum of Natural History, British Museum, and other great museums are free. In Paris, entrance to the Louvre and several other museums is free for kids under 18; also, the Louvre has a free day every month, and several small museums are free for all.
In cities such as Rome or Venice, simply strolling through the streets is an immersion in art and architectural beauty. Stroll around, eat ice cream, visit the Trevi Fountain, the (free) Pantheon: what more do you need?