There are no missing persons cases from the state of Arizona involving a woman named Mary (at least that I could find), so it seems this woman either lied about her name, was not a resident of Arizona at the time of her death, or was never reported missing.
I went through the entire Doe network looking for missing women born between 1927 and 1942 who stood 5'4 with hazel or brown/green eyes. These are all the women I came up with:
Patsy Clark (
http://www.doenetwork.org/cases/2879dfar.html) - she's slightly too short and slightly too heavy, and her eyes were green, not hazel, but all of the demographic information on her page seems to have been estimated. She was known to travel around the southern USA, Arkansas in particular, by herself, without telling anyone where she was going. Went missing in March of 1987. There is some resemblance between her photos and our unidentified lady.
Iva Edwards (
http://www.doenetwork.org/cases/3024dffl.html) - Slightly too short. Missing from Florida since 1974. Husband claims she ran off one day. Doesn't seem overly likely, since this woman is believed to have been murdered in 1974, but still, there is some resemblance.
Saunders Rymer (
http://www.doenetwork.org/cases/1067dffl.html) - Slightly too old and heavy. Went missing somewhere between Georgia and Florida in 1977.
Delores Griffin (
http://www.doenetwork.org/cases/1019dfil.html) - included because she fits the criterion, but she was almost certainly murdered by her husband in 1975.
Sharon Pinegar (
http://www.doenetwork.org/cases/1898dfia.html) - slightly too tall, and doesn't much look like our unidentified lady. Went missing somewhere between Texas and Florida in early 1987.
Senia Mattson (
http://www.doenetwork.org/cases/3052dfmi.html) - included because she matches criteria, but probably died in an accident in the woods in Michigan in 1983.
Peggy Russness (
http://www.doenetwork.org/cases/2706dfmn.html) - meets criteria, but seems to have been involved in a fire in Minnesota in 1976.
Linda Lovell (
http://www.doenetwork.org/cases/2999dfmt.html) - slighly too young. Last seen hitchhiking out of Missoula, Montana in 1974.
April Schell (
http://www.doenetwork.org/cases/1396dfnv.html) - slightly too short and light. Known to go by a wide variety of false names. Was last seen in Las Vegas in 1986. Not a lot of other information about her disappearance.
Lola Carney (
http://www.doenetwork.org/cases/1289dfnj.html) - Meets criteria, though eyes are listed as brown. Vanished from New Jersey in 1982 with $17,000 and a handgun, neither her nor her car has been seen since.
Donna Michelanko (
http://www.doenetwork.org/cases/2443dfnd.html) - meets criteria, but it seems like a bit of stretch; this woman has been missing from North Dakota since 1968.
Darcy Brown (
http://www.doenetwork.org/cases/2926dfoh.html) - slightly too short. Also seems like a stretch, since she's been missing from Ohio since 1964. Her husband's second wife also turned up missing, so it seems more likely that this woman was murdered.
Anna Rankin (
http://www.doenetwork.org/cases/2363dfoh.html) - meets critera. Was a known substance abuser and was suspected of having bipolar disorder; she was known to disappear for days or weeks at a time without contacting anybody. Was last seen in 1982, when she told her family she was moving to Florida to be with a lover.
Barbara Frame (
http://www.doenetwork.org/cases/3423dfoh.html) - slightly too young. Left her home in Ohio in 1985 and was never heard from again, though her car was found the next day.
Andrea Coyle (
http://www.doenetwork.org/cases/2495dfpa.html) - slightly too young. Had known mental health issues, and had gone missing on a previous occasion. Was known to take off and hitchhike across the country. Has been missing from Pennsylvania since 1978.
Jeanne Scrima (
http://www.doenetwork.org/cases/947dfpa.html) - slightly too short. Has been missing from Pennsylvania since 1980, though it appears very likely that she was murdered on the day she went missing.
Agnes Young (
http://www.doenetwork.org/cases/1877dfpa.html) - meets criteria. Hard to say whether or not she's a likely candidate - no one even bothered to figure out that she was missing until 2002. Known mental health issues. Someone filed a missing persons report on her in Pennsylvania back in 1986.
Sue Bean (
http://www.doenetwork.org/cases/837dftx.html) - meets criteria, missing from Texas since 1985. Not a lot of information about her disappearance.
Darline Vaughan (
http://www.doenetwork.org/cases/2896dfva.html) - slightly too short. Walked away from her house in Vermont in 1986 and never came home.
Audrey Backeberg (
http://www.doenetwork.org/cases/163dfwi.html) - slightly too tall. Audrey was known to have fled from her abusive husband in Wisconsin in 1962, taking a Greyhound to Indiana. She stated that she did not want to return.
Carol Pierce (
http://www.doenetwork.org/cases/1524dfwi.html) - slightly too tall. Husband reports that he returned to their home in Wisconsin after work one day in 1975, and she wasn't there. All of her possessions were left at the house.
Hilda Frazier (
http://www.doenetwork.org/cases/3439dfwy.html) - meets criterion. Disappeared from Wyoming in 1990. No other information on her disappearance is available.
So either our mystery woman might be one of these, or maybe no one ever reported her missing. Of all the women I found, Patsy Clark seems to be the most likely candidate.