Consider how the family views the ring's worth beyond the metal. Is its value mostly sentimental to them? Telling them the truth honors that memory better than taking money for work on a material they might not value. Would they feel cheated later if they found out, making a family heirloom into a sore spot?
But hold on, is this really such a huge deal? What if the family just sees the ring as a pretty thing to wear, not some sacred keepsake. They might be totally fine with the repair and never ask what metal was used. Spilling the beans could turn a simple fix into a big drama about honesty. Sometimes not making a big deal out of it keeps the peace.