I don't know the street parking rules throughout Staten Island, but i would be very surprised if you would be able to leave your car for a week... typically there is street cleaning 1-2 days a week on each side, so you get what is called "alternate side of the street parking" which means that you need to move your car each time the street cleaning will be on the side of the street that you have parked on. It may be less frequent in the residential areas, but this is not going to be near the Ferry (so you would therefore need to also factor into your budget getting from your car to the ferry terminal), and the city is not 'stupid' (although dealing with this at times you can easily think otherwise) and they don't want their streets used for longterm (or even daily) parking while people commute to/from work. There is a new garage near the ferry terminal that looks like it is only $8/day, but it's unattended meter facility (you pay at a pay station and then display a paper on your dashboard indicating how long you have paid for) so i don't know if you can pay for 3-4 days at a time. Either way i think you're causing yourself and family more trouble than it's worth.
You could park in Harrison NJ (or Newark NJ) and leave your car there for the duration of your stay for maybe $15/day, but you're then also adding the expense of RT PATH fare which will substantially reduce your savings on a short trip like this.
Staying in NJ is an option but the best and most convenient choices would be Hoboken and some parts of Jersey City, but you'll pay $25-$35/day to park here... less convenient but do-able is to commute by bus and stay in some parts of the Secaucus / North Bergen area, where parking is typically free, but it's a 15-25 commute (by bus) into Manhattan.
Staying (or even parking) in Long Island City isn't a viable option as you would then need to cross thru (or go around) Manhattan and you don't want to do this once (let alone twice considering you have to do it o your way home as well).
If you were traveling for a longer period (for example 7 days) there are other options where you could park at a train station in NJ and take the train the rest of the way into Manhattan... parking is often still paid but much less than NYC, and you save on gas, but the roundtrip train tickets will negate much of the savings on a short 3 nite trip
One final option is to drive yourself to Binghamton (i don't know parking options there) and take MegaBus directly to the city. Their schedule and pricing is not yet available for August, but looking into the end of June you could travel for about $7-$15 per person each way (they price by demand so the early you can book your tickets typically the cheaper it is)... this saves you on gas, tolls, wear and tear on your vehicle, and parking costs.
Otherwise, as long as you don't need to be directly near your hotel, parking in Manhattan can likely be had for about $25-$30/day.
Let us know your thoughts and we'll take it from there.