Her body was discovered in an area covered in nettles, which has led police to conclude it was implausible that she had gone there to take her own life.
A study of Mrs Brown's character and history has also led officers to the belief that she had no plans to commit suicide.
The case is set to be featured on tonight's edition of BBC1's Crimewatch at 9pm, complete with reconstructions staged in the Capital and the Longtown area of Cumbria, where her body was found on January 18.
Police believe Mrs Brown caught an X95 bus from Edinburgh bus station to Longtown on Friday, May 28 last year to visit her family. The condition of the body has left forensic experts unable to give a cause of death, but they believe it "unlikely" she died naturally.
Mrs Brown's daughter Sarah Smith and sister Maureen McLauchlan will appear on the show, where they will give moving accounts of how her death has affected their lives.
Detective Chief Inspector Jeff Ashton, who is leading the inquiry, said: "There has always been a level of suspicion, but it's been difficult for a number of reasons to draw definitive conclusions. But from looking at the scene, and at what kind of person Betty was, we are treating her death as suspicious.
"There is the fact that her bags were packed for a weekend, she had withdrawn money from the bank, and she was looking forward to being a grandmother again. That's not the behaviour of anyone planning to take their own life.
"But there are still big gaps in our knowledge, such as how Betty got from Edinburgh to Longtown without anyone seeing her. That's a gap we need to fill and we hope that the Crimewatch programme can help."
During an interview filmed for tonight's programme, Mrs Brown's daughter, Sarah, said: "My mum was a very fun, loving person. She liked her hobbies, going out walking and swimming, just an all-in-all lovely person really. She loved her grandchildren, she was always there when her grandchildren were born.
"We just want to lay her to rest, you know, so that she's in peace."
Mrs Brown's sister, Maureen, added: "She was just my sister Betty, and I loved her, really loved her. And I just miss her so much.
"I can't imagine her walking from Longtown, where she would have got off the bus, down to that area on her own. You go through all the scenarios in your mind."I just feel there's something happened somewhere along the lines."
Mrs Brown, a 55-year-old mother-of-three from Gorgie, was last seen getting off the No 3 bus in Nicolson Street. Shortly before noon, cash was withdrawn from her account at the RBS ATM on Nicolson Street.