A fire was started when the building was broken into sometime between Tuesday evening and Wednesday morning.
The museum is located in the Bogside, close to where some of the 14 victims of Bloody Sunday were killed in 1972.
John Kelly, whose brother, Michael, was killed said the damage to the building could have been a lot worse.
"I looked at it and thought: "Good God, if that had really caught fire, we could have had an explosion".
A Â£2.4m reconstruction of the Glenfada Park museum began in 2015 and is now close to completion.
Mr Kelly has been working on the project for the past 10 years.
"It shows total disrespect for my brother and all those who died during Bloody Sunday and all the others who lost their lives in that period of time, total disrespect for our family members and their memory," he said.
"The police told me there was a fire and it was in the back store, that's where we have the gas mains and electronic equipment as well.
"The building is next door to people's houses and everyone could have been affected by it. It didn't travel any further, thank God, and we still have an intact building," Mr Kelly added.
Building site foreman Dermott McGrotty said they have now improved security at the site.
"They had to climb over an eight foot fence to get in, so overnight security has now been arranged and we're making sure the doors are well secured."
The Northern Ireland Fire Service is treating the fire as deliberate. The police have appealed for information.

The 27-year-old judo athlete from Inverness was placed in an induced coma after suffering head injuries in the motorbike taxi accident on 10 May.
Since being flown to a hospital in Thailand, doctors have begun the process of waking her from the coma.
On Sunday, she opened her left eye and followed a light shown into it.
Her family joked in a Facebook post that Ms Inglis opened her eye in shock at hearing her father, Robert, singing by her bedside.
In the same message, her family added that her temperature was falling and had almost returned to normal, an indication that she was fighting infections, including pneumonia.
Ms Inglis had been in Vietnam for four months teaching English to underprivileged children when the accident happened.
It is understood that a long skirt she was wearing, and which she had tied up for the bike journey, became unravelled and caught in the wheel of the motorbike, pulling her off the machine and on to the road.

Hilary won the Theakstons Old Peculier Crime Novel of the Year Award in Harrogate on Thursday.
Radio Times TV editor Alison Graham, who was one of the judges, described the book as "an emotionally exhausting and powerful story".
US author Sara Paretsky, who created detective VI Warshawski, received an outstanding contribution honour.
The accolades were handed out at the start of the North Yorkshire town's annual crime writing festival.
Shetland and Vera writer Ann Cleeves, this year's festival programming chair, said the crime novel of the year shortlist had been "really strong" this year.
The panel had felt Hilary's winning book was "so finely written and tightly written", she said.
Someone Else's Skin, a thriller that tackles domestic violence and so-called honour crime, is the first in a series of novels featuring Det Insp Marnie Rome.
"The author was able to conjure up atmosphere in very few lines," Cleeves said. "The subject matter was well done and well plotted."
The other books on the shortlist were:
Sara Paretsky was chosen for the outstanding contribution to crime fiction award after her 23-year career writing about VI Warshawski, a female private investigator from Chicago.
Actress Kathleen Turner played VI Warshawski in a 1991 movie based on the Deadlock novel and in a BBC Radio 4 adaptation.
VI Warshawski was "a real game changer", Cleeves said. "This was a woman who didn't just solve crimes but was like an action hero in a sense."
Paretsky "really changed the way that readers thought about female writers", Cleeves said.
"When she started writing, it was pretty well unheard of to have a strong female protagonist," she added.
Paretsky also set up Sisters In Crime, an organisation to support female crime writers around the world.
An arson attack on the site of the new Museum of Free Derry shows "total disrespect" for victims, a Bloody Sunday relative has said.
[NEXT_CONCEPT]
Commonwealth Games silver medallist Stephanie Inglis has opened an eye for the first time during her recovery from a serious accident in Vietnam.
[NEXT_CONCEPT]
Author Sarah Hilary has won one of the UK's top crime-writing awards for her debut novel, Someone Else's Skin.