The area around Times Square in Manhattan has lots of hotels; have a look too on tripadvisor's website.
Would suggest you visit places like the Rockefeller Centre (Top of the Rock can have fine views), Empire State Building, Central Park, WTC site and South Street Seaport during your time there.
Child prices for airfares will no longer apply; you will need to pay adult fares for the children. Do make your children aware too that they will be fingerprinted and photographed as well as yourselves on arrival into the US and they will have to comply. Also you will need to go online once you have booked the flights and accommodation to complete the ESTA form four times in succession (one form is needed for each of you). This currently costs $14 US per person and is valid for 2 years from date of application. Use only the official US government website to do this - its https://esta.cbp.dhs.gov, use no other as they are unofficial and overcharge people.
New York is a bit further than the Med, hence the price difference. Personally I would go because it is a fabulous city and everyone should experience it. You will get jet lag so go early in the week. Personally I would consider going at the end of August which can be cheaper. New York empties in August and it can be hot but hotel deals are better. Also, there is no such thing as a late cheap deal to New York. Go on the BA site and check out the prices in late August. Booking a long way ahead should be better value.
For ease of walking, I would stay centrally in Manhattan. Although Brooklyn is now fashionable, the commute will have to be factored in. I would add the Highline to sightseeing musts and also MoMa and the Guggenheim. 5th Avenue is great and so is Central Park. Take the Staten Island ferry for the best views of the Statue of Liberty. Go to Ellis Island where the immigrants had to wait for approval for entry. Some really famous people landed there. Eating out can be expensive. Do not take breakfast in the hotel, go to a local deli type cafe. There is so much to see working out what suits you best is the difficult part, but do go.
Talkin thanks (and everyone else, of course) but- why's Air bnb (who I hadn't heard of before now!) banned in the USA?- not that I'm expecting you to be an expert- but how can you make your payment secure and binding if you use such folk? (Only ever time my CC was ripped was using it from the UK to buy something for the US!)
tansie you can still book through AirBnb but look at Jersey City or Union City on the other side of the Hudson. It is New Jersey state, New York legislation doesn't apply! 10-20 minutes on either the Path train or bus to Manhatten. We've just booked a lovely flat in Jersey City for October half term, £50 per night for 3 of us.
Unfortunately the days of cheap airfares to the US are long gone; the rising cost of fuel and APD (Air Passenger Duty) from the UK which is in effect a tax on passengers imposed by the government, has caused airfares to rise generally.
Short term lettings of apartments in NYC is illegal and many building owners do not allow this to happen at all. (Also many residents do not like tourists coming and going from their building all the time). Many people have tried renting apartments there and have been scammed for their troubles so do not take the risk of actually renting an apartment in NYC. Its too much of a risk and the apartment you've booked and paid for may well not exist or at least be the same as advertised.