Bäck is a very common village name, and since there are three parishes named Tveta (one in each of Stockholm, Kalmar, and Värmland counties) I'm not even sure I have found the right Bäck. I'm assuming here that you are referring to Tveta in Värmland county, since that's the only parish with a village named Bäck that I can find in the Rosenberg gazetteer.
Rosenberg is very brief here: Bäck in Vermland county, Näs hundred, Tveta parish. Village.
Fortunately, I was able to locate Bäck on sheet 10C Åmål SO of the 1:50,000 scale topographical Gröna kartan map series. It can be found in the north-western part of Tveta parish, west of lake Sjönsjö, along the road between Kila church and the city of Åmål, just 1-2 km north of the border between Älvsborg (now Västra Götaland) and Värmland counties. In case you have use for longitude and latitude, it's at 12?45'E 59?09'N. Maybe someone familiar with the lmv.se website can show a map image.
With no description in Rosenberg and only five small dots (houses) on the map, I can tell no more about the place. As for the name, the pronounciations of short e and short ä are very similar, and spelling has changed in general (usually from e to ä) of many names and words over the past century. Even if Bäck happened to be the common spelling already 100 years ago, it could be that Sara considered the spellings to be equivalent enough for her to pick Beckman for her surname (especially as that would make it easier to write in English). Today the spellings Beckman and Bäckman seem to be about equally frequent in Sweden.
The Swedish word bäck means brook (small creek).