To reduce cost and complexity you have to make a decision on what chips are included in your design, for example only 5V supply range, only HCMOS families. The IC under test itself goes in an external zero insertion force (ZIF) socket that can take, for example, 14 pin or 16 pin narrow (0.3 inch) DIL chips. It would be a step more to have wide DIL packs (0.6 inch) up to 24 pin or more. Surface mount? Exclude them. This is an exercise in design to show you can do it, I expect.
Another decision is whether specialised chips like timers, ADC, DAC or analogue switches will be included. These need additional input types, output types, power supplies and test routines. Leave them out. These are all mixed analogue and digital.
The inputs have several possible states, high, low, pulses (clock inputs for sequential devices like counters and flip flops and shift registers), or no connection.
The outputs have several states, like high, low or tristate or open collector or sequential (as in a counter or shift register or flip flop).
The power pins are just two types for a basic tester, 5V and gnd, only changing for a 14 pin or 16 pin device. A single relay with 2 changeover contacts would suit this switching.
The test involves programming the input pins and output pins to connect to the appropriate states, and running through the truth table for the device. There is only a limited number of standard logic chips, probably 100 or so, and most of the tests are similar, simply changing the truth table and pins used. You will need to figure out how to deal with counters and shift registers and some flip flops, which have outputs that depend on the previous states (sequential). The display can track the states of outputs compared to9 inputs, by remembering previous states.
The simplest method would be manual jumpers or switches to connect the pins as inputs or outputs according to the data sheet, and visually observe the output states (perhaps displayed on leds), as inputs are switched through the combinations of the truth table.
Greater sophistication would be to test the inputs and outputs for specified threshold levels, and to program the connections of inputs and outputs to pins automatically, using software with a list of all the standard IC types. The truth table tests could also be automated. A further level would be to measure the rise times, fall times, propagation delays, maximum clock rates, effect of loading outputs etc. All the required information is available on data sheets. I would think though, that just testing the truth tables, perhaps with threshold values, full loading of outputs, is sufficient, as a faulty device will usually fail that. One other useful test is the total current drawn by the chip.
The connections to the pins might use relays or analogue switches. These could be controlled by a micro-controller connected to a PC for example, with a serial or USB interface to the PC if you can program a PC. One controller device you could consider is a Maximite, which is less than 100 dollars, has 20 spare I/O pins, and could act as a stand alone system (no PC) as it can have a normal PC keyboard and video display connected. It also has plenty of file storage for the device tables using SD flash cards. The I/O pins of the Maximite can be expanded with standard decoder chips, say 4 lines to 16 lines. There is no easier system to use for this. One point - the Maximite is 3.3V power lines and I/O, but that is easy to deal with. Perhaps stipulate only HCMOS, which can also work at the same voltage level. You can just connect 14 of the i/o lines to the pins of the ZIF socket (IC) under test, and program them as inputs and outputs. Another I/O line connects the power and ground to the right pins.
Make sure the software allows you to enter a new device type by a simple fill-in-the form method, and you don't have to have all the known devices installed. Just a few representative demonstration ones. like a counter, various gates, a decoder or encoder.
The third link shows an 18 pin ZIF socket that would also suit 14 and 16 pin devices. There only needs to be one of these.