One thing my kids enjoyed was having a kids' National Park "passport". It's available at eparks.com and at many national parks and monuments, the kids can get their "passport" stamped for free (there is a small charge for the book). It's a nice way to get a little memento without spending tons of money at the gift shops. The website has a list of which national parks and monuments that give the kids their stamps on their book. Usually the stamps are given at the entrance, or the guards' gates, or the office.
When we have been in DC and similar places, instead of going overboard (if money is a concern) at gift shops, we let the kids get their books stamped, and let them get the flattened, embossed, souvenir pennies (cost: 51 cents). We got them penny keepers (available online) to keep the pennies in. We also had fun looking through change for quarters that depicted the state that we were visiting. Then near the end of trips, we allowed them one take-home small gift. Don't forget that things bought at gift shops have to be lugged around for the day. A kids' passport and a penny is easy to take along. You can purchase the passport booklets and the penny binders in advance.
I agree that the Jefferson and Lincoln monuments are worth seeing. Your kids will see them for years to come in commercials, movies, and in photos, and they'll remember being there in real life, and how big and impressive those monuments actually are.
What are your kids' interests? Don't try to do it all, or rush. Think about what your kids enjoy. Try to balance history and current events (government buildings, Vietnam Wall, for example) with fun (the zoo or the Air and Space museum, etc).
If you'll be taking the Metro, this map shows hotels near Metro stations:
http://hotelsneardcmetro.com/hotels-in-washington-dc-near...
Sometimes it's good to balance budget with convenience. For example, you might find a really great hotel price, but the hotel is so far out of the city that you need to spend lots of money on cabs in order to get to the Metro or to go downtown, and you waste lots of time driving in unfamiliar territory, and lots of money on taxis. Paying a little more for a hotel that is right near the Metro means you can easily get around and get back to your room quicker, and it means less walking for tired children!