Cleaning with sodium bicarbonate
If originally made of silver, the dark layer on your jewlery is silver sulfide, $\ce{Ag2S}$.  The removal suggested by your mother is an electrochemical one, that will reduce the silver back into metallic, shiny $\ce{Ag}$, if in addition of hot water and bicarbonate, salt and an other metal acting as reducing component is in electrical contact with the silver piece are present.  Here, some foil of aluminum will serve well as sacrifical anode.

To quote UMass Amherst, presenting it as a lecture demonstration, the chemistry may be summed up to this equation

$$\ce{3 Ag2S(s) + 2 Al(s) + 3 H2O (l) –&gt; 6 Ag(s) + 3 H2S (aq) + Al2O3(s)   }$$

The addition of salt (which is absent in the equation) serves to increase the electric conductivity of the solution, hence increasing the rate of this process.  Immerse the jewlery in the hot solution (it needn't be boiling), for example wrapped in the creased foil for some minutes; then rinse and wipe it with a cloth.