Having installed POS and phone systems, I can give you some information from direct experience...
The NEC DSX-80 will do this - you can install it yourself or have it professionally installed.
How many tables plus other locations for this closed phone system do you want? (E.g., 17 tables + 1 host + 1 kitchen = 19 stations)
In this example, you'd get a NEC DSX-80 phone system and install a single 16 port analog station card, and a single 8 port analog station card giving the system 24 ports of capacity (since in this example 19 ports are needed and cards come in 16's and 8 port models).
It also comes with a card called a 8 port trunk card - you just put this card to the side and do not connect it (trunks connect to the phone company).
Each table would get an extension number like 316, 317, 318, etc.
You can run the cable and any standard phone you get at the local office store will connect to the analog station card. If the phones have a caller ID display, they'll tell the ringing station which table is calling them.
A waiter, the host, and the kitchen can have phones. You can even give them cordless phones. If you give them a cordless phone with caller ID, they can tell which table is calling.
The system also comes with a "digital station card" which is for connecting NEC phones. These phones cost a little more. You'd only need 1 of these phones to program the system.
The tricky part - and VERY EXPENSIVE part - is getting the billing you want it place. Your point of sale system (POS) will actually need to be interfaced to the phone system. The problem is, I've never seen a POS system (e.g., Micros, Digital dining, and the like) that will interface to a phone system. We've installed property management systems (PMS) in hotels where the phone system interfaces to the PMS and the POS interfaces to the PMS, but not where the phone system interfaces directly to the POS.
You're almost better off saying "our tables have phones, we can turn it on for $5" and having a patch panel at the host's podium where they can connect and disconnect a patch cord to turn the phone on or off. A custom software package could cost THOUSANDS and THOUSANDS of dollars.
A system WITHOUT the billing software will be SO inexpensive - possibly "cheap" even compared to a system with a billing system that the novelty of having it - for free to your customers - may sell more food than minutes. That's our guess.
The phone system DOES have the port called a SMDR (Station message detail report) port. The SMDR port is the port where you call accounting (billing) software connect to the phone system - these are very standard ports - the trick is finding an affordable call accounting package that doesn't "get in the way" of your primary business which is selling food. There are some cheap call account packages, but you don't want to have to run a station report every time someone is done eating! This is why interfacing the phone system directly to your POS is important.
To sum it up -
It can be done using the NEC DSX-80.
It can be done without billing software for a reasonable price.
It can be done with billing software for an astronomical price.