Ouch. I would never take money without a signed contract in hand first. I know that doesn't help now.....If it wasn't in writing they may have a case...but I would tell her per your cancellation policy that they lose the deposit and go from there. Why in the world were they renting your home if they are local and own an inn? Something doesn't seem right. Also, it might not hurt to go back to everyone that inquired and tell them the property has become available. Maybe you can catch some that haven't found anything yet. Good luck!
Our policy states that"in order to secure my home, we require a non-refundable deposit (part of the total rental fee) of $1000". We ask our guests to be very sure that they wish to rent my home as the deposit is never refunded. That is not to say that we have not refunded on a few occassions. Young couple lost their jobs 3 weeks apart and could no longer afford to rent, a father who was the sole support of a family dies unexpectedly, etc.
Make sure this is stated on the Homeaway site as well as your own wesite. That way if you do not receive a signed contract back, they were informed.
Refund the $1000 in full with a cover letter explaining that their reservation has been canceled and you are sorry that they did not understand that the rental rate is $2400 per week (is the $3500 normal rate published anywhere? If so, include a printout of it with the letter).
Even if they want to question the amount you verbally you agreed on - arguing that it should be $2400 for the month - your position is pretty strong, in that you did not even receive the 50% of that amount (i.e., $1200).