Thulani Maseko was shot dead on Saturday night by unknown attackers in Luhleko, around 50 kilometers (30 miles) from the capital Mbabane, opposition spokesman Sikelela Dlamini said.
The spokesman said he was told "assassins shot him through the window while he was inside (the) house with his family".
"Details are still scant (and), owing to the trauma his family members are undergoing, they are not yet ready to speak," Dlamini added.
The government sent condolences to the family, saying Maseko's death was a "loss for the nation" and that police were searching for the killers.
Maseko was a leading human rights lawyer and columnist in Eswatini who had a pending court battle with King Mswati III over the monarch's decision to rename the country Eswatini by decree.
The country's name was changed from Swaziland to Eswatini to mark the 50th anniversary of its independence from Britain in 2018.
Maseko's position was that the king had not followed the constitution in the process.
The European Union called for the killers to be caught and voiced "grave concern" about the situation in Eswatini.
"The EU calls on the authorities to ensure the safety of all citizens, including political activists," it said in a statement.
The U.S. Embassy at Mbabane expressed "profound sadness" and extended "deepest condolences to Mr. Maseko's family, friends and admirers around the world".
"Eswatini and the world have lost a powerful voice for non violence and human rights," the embassy added.
Last week, the Swaziland Solidarity Network (SSN) alleged that the king had hired mercenaries, mainly white Afrikaners from neighboring South Africa, to help Eswatini's security forces suppress rising opposition to his regime.
But government spokesman Alpheous Nxumalo said "no hitmen have been hired".
Rights group Freedom Under Law, which operates across southern Africa, pointed a finger at the government.
"Somehow the stunning news that Thulani Maseko has been gunned down in cold blood comes as no surprise," it said in a statement.
"A ceaseless and fearless human-rights lawyer, an outspoken critic of the regime in his beloved Eswatini, Thulani had all too long suffered at the hands of a heedless regime."