I appealed based on what previous years have suggested once I was awarded an initial amount of financial aid. Past forums haven't been the best about updating whether or not this has worked for them (people fall off posting once they commit), but I'll try my best to let you know if anything comes out of my various appeals. Here's the basic formula I used for Columbia:
1. Explain to them that their school is your top choice due to x, y, z
2. Thank them graciously for the amount they have offered.
3. Indicate that your choice in commitment will ultimately come down to funding. If you've been awarded a substantial amount to somewhere else: "another top program has awarded me x amount, which will reduce my yearly tuition to y amount." You could also bring up your income here (I did) or cost of living differences.
4. Don't ask directly about matching an offer, I said something along the lines of "avenues of alternative funding" or something like that. Ask straight up for the appeal though!
Remember that you've worked really hard to get where you are, and you deserve access to higher education.