Mrs Brown, who should have been celebrating her 56th birthday today, was last seen on CCTV getting off a bus in Nicolson Street at 11.55am on May 28.
Relatives have been left increasingly anxious about the fate of the missing mother-of-three, making repeated appeals for her to get in touch.
Today, her sister Maureen McLaughlin said the family would "dearly love" to be reunited with Betty, from Gorgie, on this special day.
"It is Betty's birthday, and we would dearly love it if she was here to celebrate it with all of her family, who still miss her terribly.
"It is now over five months since she was last seen, and in that time we have grown ever more anxious for any information that can help us find out where she has gone," she said.
"Everyone misses her, and we are desperate for any news that can help put our minds at rest. If anyone has any information that can help, then please get in touch with police.
"Betty, if you are reading this we want to wish you a happy birthday, and let you know that we still miss you so much.
"If you don't want to contact us then please let the police or somebody else know that you are ok. All we want to know is that you are safe and well."
It is understood there are no fresh leads on the inquiry and Chief Inspector Murray Dykes, who is leading the investigation, said there had been no confirmed sightings since she went missing.
"Betty's birthday should be an occasion for celebration, but instead we are appealing for information that can help us reunite her with her family," he said.
"We have devoted significant resources towards tracing Betty, however as yet we have had no confirmed sightings."
In September, police investigated a possible sighting of the missing woman outside Sainsbury in Deanhaugh Street, Stockbridge, following a call from a member of the public.
The family believed there was a genuine likeness to Mrs Brown, however it was later ruled out.
Mrs Brown was reported missing after failing to return to work at a care home in Gilmerton following a week's holiday.
Also known by the surname Hunt, she was last seen wearing a black jacket, open-neck blouse, blue jeans and black shoes, and was carrying two large Aldi carrier bags.
She is described as around 5ft 7in, with fair hair, and glasses.
Originally from Hawick, police believe she may have travelled to her home town from Nicolson Street, but sources say her disappearance is out of character. Her bank cards have been unused and her mobile has remained switched off.