Originally Posted by jib71

Flat fares are automatically applied for rides from HND but it may be that fixed rate won't apply for your trip. I looked at Nihon Kotsu and KM Taxi's websites - They have maps that show the fixed fares to various city wards. But for the closest areas, like Minato Ku, where the Conrad is located, meter fare applies. I would not expect it to cost more than the fixed fare to wards that are further away. You can expect traffic to be light at that time in the morning.

Be aware that cab fares are 20% more expensive from 10:00 PM to 5:00 AM. If your flight lands super-early and you take a cab just before 5:00AM, the elevated rate will apply until 5:00 and then the normal rate will kick in.

EDIT - Although you're taking a cab, you should buy "Welcome SUICA" or "PASMO Passport" cards for yourself and your family at the airport. (If you forget at the airport, you can buy PASMO passport cards at Shinagawa station near the Conrad https://www.pasmo.co.jp/visitors/en/buy/). The cards will help you to use public transportation as much as possible for travel around Tokyo. Be aware that they expire after 28 days, so you should use up the balance during your stay.