The boy's mother, who wishes to remain anonymous, contacted the museum after reading about the appeal online.
The Delftware puzzle jug had broken into 65 pieces.
Original versions of the jug can fetch thousand of pounds at auction.
The boy was thought to be about five when the accident happened last summer.
A spokesman for Ipswich Borough Council said: "The appeal to find the mystery boy attracted media interest not only here in Suffolk, but throughout the UK and even America and South Africa.
"We received a call from his mother who told us the family were delighted that the puzzle jug had been repaired but that the family wished to remain anonymous. We only know the family were visitors from somewhere in the South East."
Carole Jones, who is in charge of Ipswich's museums, said the boy knocked the 221-year-old "beautiful, ancient puzzle jug" from a low window ledge.
"He knocked it off and it smashed into about 60 pieces," she said. "He was of course, absolutely devastated, and his family were really upset."
The restored jug has been moved from Christchurch Mansion and is on display at the Ipswich Art Museum in the High Street.