burakzilla:
It basically works as setup in the attachment below, the issue is it will skip cycles randomly. When I connect just the motor and battery from the device to the relay, it works like a champ. We measured the resistance of the original membrane switch and it is ~65 ohms and the relay ~5.
By 'motor and battery do you mean you just connected a completely different circuit through the relay to prove it was closing properly?
When you say 'skip cycles' do you mean the circuit under test doesn't behave as if the 'membrane switch' had been operated, or do you mean the relay isn't closing? You can test for closure by measuring the voltage difference across the switch.
If the relay is significantly lower resistance than the membrane switch it is simulating, it may affect the circuit under test. To avoid the risk, you could put a suitable resistance in series with the relay.
Have you confirmed that the coil circuit and the switched circuit inside the relay are electrically isolated from each other? They should be, but it never hurts to double check.
Is there any other electrical connection at all between the circuit under test, and the Arduino? Any shared power supply you forgot to mention, or anything like that?