Police said they received a report of the fire at a property in Castle Mews at about 03:30 BST on Saturday.
A tyre had been put against the front door of the house and set alight. The blaze caused smoke damage.
A man, a woman and two boys inside the property were not injured.
Det Insp Tara Nicholls said: "I would appeal to anyone that was in the area and noticed any suspicious activity between the times of 01:30 and 03:30 BST to contact CID at Cookstown Police Station on the non emergency number 101."

Dave and Leslie Blissett, of Lincolnshire, bought the Tornado GR1 - extensively involved in the 1991 conflict - as a shell.
For sale on Gumtree, it has no engine, or guns and the wings have been removed for transportation purposes.
The ex-RAF engineers, whose love of aeroplanes brought them together, said it would make an ideal museum display.
Mr Blissett said he has always had a passion for taking things apart.
He took on the project after being involved in the restoration of a Lightning jet at RAF Binbrook, near Market Rasen.
"What we started with was literally a bare frame - you couldn't recognise it as a Tornado," he said.
"Over the years we've bought, swapped and gone through scrap bins to get it to where it is now."
Mrs Blissett said when it first arrived it looked like a "metal caterpillar on the back of a lorry".
"Some people go for garden gnomes, we go for aeroplanes," she said.
The Panavia Tornado GR1 was used extensively in combat by the RAF in 1990s, including in the Gulf War in 1991.
On the classified ads website, the couple have written: "She is virtually complete and is ready to move as her wings have already been removed."
Andrew Panton, manager of the Lincolnshire Aviation Heritage Centre, said it was "unusual" to sell aircraft on Gumtree, as they are generally sold via the Ministry of Defence surplus service.
"They way they are doing it - making it known worldwide - could work in their favour," he said.
As interesting as the piece is, it is the "wrong era" for the centre in Spilsby, which focuses on World War II aircraft, he said.
Four people have escaped injury after an arson attack in Cookstown, County Tyrone.
[NEXT_CONCEPT]
A Gulf War-era fighter jet has been put on sale online for £20,000 by a couple who took six years to rebuild it.