The T3 Terminal is open throughout the night and you should easily be able to move around anywhere publicly accessible inside it. If you have a visa, you can just as easily go out. Since you mentioned you have a domestic flight, I'll assume you do have a visa. Unfortunately though, there is a lot of hustle and bustle due to international flight departures in the night as well. I doubt you will be able to find peace and quiet at the airport, but there are some options.
There isn't an answer which can suit everyone, but depending on your budget, there are some great options for hotels which are either inside the airport or offer a pickup service. Almost all of them offer free airport pickup service so it should be quite easy to get to them. Also, quick tip - Taxis in Delhi are not very expensive. Here's a list:
I want something at the airport! (These options DO NOT require a Visa)
Now doesn't this restrict your options a little bit. If you're not too concerned with services, wish to keep the price low just need a place to crash for a few hours, these are your best bet.
The airport website offers some insight into the facilities. It also allows you to book only a shower at T3 Arrivals if you just prefer that (About $7). Be sure to check that out as well.
5 Star Hotels - Expect anything greater than $100
These 3 five stars are more or less in the same complex and are extremely popular among business traveller. If you want good service at a decent price, check out Radisson Blu.
4 Star Hotels - Expect anything around $40 - $80
Le Seasons is in the same complex as the other five stars I mentioned, which makes it a good option. Shanti Palace is relatively better but I believe Le Seasons should offer a better experience due to the cleanliness around Aerocity. Also, it is relatively cheaper. I put Eaton on the list even though it's inside the airport, it is a legit hotel but it is absurdly expensive with sub-par service.
3 Star Hotels - Expect anything around $20 - $50
This is a tough cookie, but I would suggest either Delhi Aerocity or De Aura. Whichever is cheaper for the date you find it at. Ibis is relatively new in the area so it does have good services, but higher prices.