I am putting this comment here since it seems the most obvious one when searching for Gatwick Arrivals.
The Hampton Inn, part of the Hilton Group, has a new hotel built into LGW's North Terminal which offers day rooms for £55 a day per room. They can't be pre-booked, you turn up on the day and you will find out if they have space. The rooms are available from 09:30 hrs until 18:00 hrs. If you turn up before 10 am, you can also have the breakfast too, though I'd put the value of that at around £5 to £10 a person.
Rooms are modern, most are queen sized beds, some rooms have a sofa bed as well, and others are twin bed. The bathroom isn't that big, but has a walk in shower. There's a gym, the public areas are very attractive, with a bar, snacks for sale and a Starbucks concession. If you are not familiar with Hamptons, they site above Holiday Inn Express and below full Hiltons in terms of style and comfort, several notches above Premier Inns.
So for a couple or a family this may be reasonable value for money. The hotel is on the middle level between arrivals and depatures (same level as the South transit), near the car hire desks. It used to be Longbridge House, the admin offices of BA and staff canteen. The floorboards are still a bit creaky!
Link to the hotel's website here.