A video of the Salford taxi driver's killing appeared online on Friday.
His widow Barbara and their children said they were "extremely proud" of the 47-year-old, who had been delivering aid to Syria when he was kidnapped.
They thanked the government for its support although Mr Henning's brother-in-law had earlier suggested more could have been done to secure his release.
Mrs Henning's brother, Colin Livesey, described the killers as "scum" and said the family had lost a "great person".
'Extremely proud'
Prime Minister David Cameron, meanwhile, described the murder as "completely unforgivable" and vowed the UK would do all it could to find those responsible.
Mr Henning was on his fourth aid mission to Syria in December when he was kidnapped minutes after arriving in the country.
Floral tributes have been left for the murdered volunteer in his home town of Eccles, Salford
Mr Henning was a taxi driver in the area
The statement from the Henning family said: "Alan was a decent, caring human being. His interest was in the welfare of others. He will be remembered for this and we as a family are extremely proud of him and what he achieved and the people he helped."
Mrs Henning and the couple's children, Lucy and Adam, also thanked those who campaigned for Mr Henning's release, saying: "Your efforts were a great support to us, and we take comfort in knowing how many people stood beside us in hoping for the best."
Earlier, Mr Livesey - Mrs Henning's brother - said he was devastated for the family.
"We're all just saddened knowing that we've lost a great person in our family," he said.
Regarding Mr Henning's killers, Mr Livesey said: "I just hope and pray they get what's coming to them. I've just so much hatred for them."
He said the government could have done more for Mr Henning in captivity, "when they knew about it, months and months ago".
'Absolutely abhorrent'
The prime minister described Mr Henning as a "kind, gentle, compassionate and caring man".
Speaking at Chequers following a briefing from defence and intelligence officials, including the head of MI5, Mr Cameron said the murder was "absolutely abhorrent".
"Anyone in any doubt about this organisation can now see how truly repulsive and barbaric it is," he said.
A Downing Street spokesperson added: "Isil's brutality will not persuade us to change our approach.
"Indeed, the senseless murder of an innocent man only reinforces our resolve to defeat this terrorist organisation and to eradicate the threat they pose to Britons - whether those in the region or here on the streets of the UK."
Alan Henning was on his fourth aid mission to Syria
In the most recent video published by IS, the extremist group sometimes known as Isil or Isis, threatens another aid worker - American Peter Kassig.
BBC security correspondent Gordon Corera said the footage included a reference to last week's vote by Parliament to authorise air strikes against the militants in Iraq.
As in previous IS videos, it features a jihadist who, from his accent, appears to be from Britain.
Mr Cameron said "all the assets" the government had would be used to try to find and help remaining hostages.
A Downing Street spokesman also said police were investigating a second video which surfaced on Friday night.
It shows a British IS fighter, without his face covered, delivering a further message.
Kasim Jameel, the leader of the aid convoy Mr Henning was part of when he was taken, said he had believed his friend would be released and told the BBC he was "distraught" by his death.
"The captors knew that he was just an aid worker," he said. "This was just a game for them. They knew what Alan was about. I thought getting to know Alan for such a long time they would release him."
He said he hoped to set up a charity in Mr Henning's memory so "he is not forgotten".
Analysis
Frank Gardner, security correspondent
David Cameron's vow to catch the jihadist killers of Alan Henning and bring them to justice would seem to match the mood of the nation.
But judging by the track record of previous such cases of hostages being murdered overseas, this promise has little likelihood of being fulfilled.
Tony Blair made the same pledge after Ken Bigley from Liverpool was beheaded in 2004, Gordon Brown did the same when tourist Edwin Dwyer was kidnapped and killed in the Sahara, and Mr Cameron vowed to punish those who besieged the Algerian gas plant last year.
According to the Crown Prosecution Service, there has not been one single case of any murderers of British hostages ever being brought to justice in Britain.
Profile: Alan Henning
Henning's home town 'stunned'
'Generous character'
IS has previously released videos showing the apparent beheading of two US journalists, James Foley and Steven Sotloff, and British aid worker Mr Haines.
The video released on Friday is yet to be verified, but it appears to show Mr Henning kneeling beside a militant dressed in black, in a desert setting.
The footage ends with an IS fighter threatening a man they identify as Mr Kassig.
In a statement, Mr Kassig's family said he had converted to Islam. They referred to him as Abdul Rahman Kassig.
The family asked people to pray that he and "all innocent people being held hostage in the Middle East and around the globe" would be freed.
John Cantlie was kidnapped in Syria in 2012
Earlier on Friday, the father of another hostage, British journalist John Cantlie, appealed for him to be released "to those he loves and who love him".
The journalist, who was kidnapped in Syria in 2012, has so far appeared in three videos in which he has delivered scripted messages responding to military attacks on IS.
RAF Tornados first hit IS targets on Tuesday, four days after Parliament authorised UK involvement in an international military campaign.
The aircraft have been conducting daily flights over Iraq, and carrying out air strikes against vehicles and weapons positions to assist Kurdish and Iraqi army ground forces.
IS has seized large parts of Iraq and Syria and declared what it describes as a caliphate in the areas it has taken.
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