In NY, the executor to the estate is in charge of the distribution of the items in the will and makes sure that the estate is assigned a Tax ID number (the lawyer will actually do the paperwork). Your responsibility is to essentially make sure that the checking account is open (need the Tax ID #) and that all bills are paid from that account before the money is distributed to those mentioned in the will. You are basically the CEO/CFO for the "account" until it is settled. Most estates take 6-9 months to settle with respect to the finances.
If there are items in the estate (usually personal items or small household items) that are not specifically left to a person or a group, the executor is responsible for either arranging the estate sale or donation and making sure that the $$ received from the sale is divided appropriately according to the will.
You are not personally responsible for the debts of the individual (hence the separate ID # and checking account), but you will be responsible for making sure that all of the debts are paid on time and that the will is dispensed per the document.
My mother is the executress to my grandfather's estate and this is pretty much what she did after his death. Depending on the complexity of the estate it can be a lot of time in the first few months following the death. My mother lives out-of-state and had all bills and mail forwarded to her in FL. She did travel here (NY) twice to sign paperwork, but that money came out of the estate account b/c it was equivalent to a "business trip".
Before you make the decision, I would ask the person to see the will or to at least describe to you the complexity of his/her allocations. Make sure that the attorney who drafted the will is available to you for consult and that you have an accurate listing of the "typical" bills they pay (which credit cards, electric, water, etc) so that you don't forget to pay something or forget to have something "closed" and "cancelled".