Her son, Sam, married on March 10, 1909, in Cleveland, Ohio. See Sam and Sadie's marriage license where the groom's father is Adolf and his mother is Sali Handler.
Her son, Josef, married on March 10, 1909, in Bonyhád, Tolna, Hungary. See Josef and Lena's marriage record where the groom's father is Aron Handler and his mother is Szoli or Száli Handler.
Several Handler family members arrived in New York City on May 25, 1911. See Handler Family passenger lists where she is listed as Roza Handler.
Her daughter, Regina, who arrived in New York City with her in May 1911, married in Cleveland, Ohio, on December 13, 1914 to Jacob Solomon. Regina's marriage license lists her father as Adolf and her mother as Rozalia Handler.
In the 1920 U.S. Federal Census, the household of Regina (Handler) and Jacob Solomon includes a 60-year-old Rose Goodman, listed as mother and having immigrated in 1911. This may be Regina's mother, as Jacob's mother's name is Anna (as listed on their marriage license). Why she is listed here with the surname of Goodman is a mystery, assuming this is Regina's mother.
The biggest mystery is that I don't know when or where Rozalia (Roza, Rose, Szali, Sali) died, and I cannot find Regina and Jacob Solomon in the 1930 U.S. Federal Census to see if Rose Goodman is still living with them. My next task is to explore Cleveland, Ohio, City Directories.
Son, Sam Handler, died in 1952. His death certificate lists his mother as Sally Handler-Yugoslavia
However, she is listed as "Don't Know" on her son Joseph's death certificate in 1947 where his father is remembered as Aaron.
I have not found the Hungarian record for her marriage or her birth. (In fact, I'm not sure where in Hungary she was born, possibly Ilok, which is now in Croatia.) Until I find additional records, I am going to suppose that her name was Rozalia, and she was recorded as Roza, Szoli, Sali, or Sally, depending on the source.
December 10, 2012 update: I found a death certificate for Rosie Goodman and a burial record at JewishGen Online Burial Registry for Sally Handler who I believe is the same person.