The 7805 is good for the Arduino.
If the output is for example 4.9 V, that is not a problem.
The Arduino will work with lower voltages, but if the output of the 7805 is lower than 4.9V, the 7805 is broken.
What kind of multimeter do you use ? Is it a cheap one that could be 10% off ?
Do you use a lot of current for something. If the 7805 is getting hot, the output voltage is reduced.
The 7805 needs 100nF close to the input pin and 100nF close to the output pin. At the output a capacitor of 22 ... 1000 uF should also be used. Also such a capacitor is needed at the input, but in some cases it can be smaller.
If you need an exact known voltage, use a voltage reference.
The microcontroller on the Arduino board has an internal voltage reference. But you can buy such components if you need higher accuracy.
If you want for example 5.05V and it doesn't need to be very accurate, you can use a DC/DC converter with a adjustable output.