I've consulted dwds.de on the day name of Saturday. It is obvious that "Samstag" prevails nowadays like "Sonnabend" prevailed 70 years before. The reason for this shift remains unknown to me.
Saying that either version prevailed in German is a broad claim, as the different usage splits German landscape for more than a millennium in a south-westen region using Samstag and a more north-eastern with Sonnabend. Not to mention that it as well goes along the split between Catholic and Protestant regions. Confession has done a lot to solidify the split since the 15th century.
A 'Why' for a change is never a hard fact, but what can be noted is that expansion or contraction of usage in recent history goes somewhat in line with political changes.
Between 1870 and 1945 Protestant Berlin with native Sonnabend had a major role in setting nation wide wording - of course with exception for Austria and Switzerland.
After 1945 and until 1990
- In Western Germany traditional western/southern influences of Samstag was not only more widely used, but moving capital and government to Catholic Bonn gave it an additional push, while
- Eastern Germany not only kept Berlin as center of power, but was almost entirely made up of areas where Sonnabend was native.
After 1990 West German norms were almost instant assumed to be the way for reunified Germany, wich included the now 'official' Samstag.
Was it in the past often a matter of congregation so took modern media that role in recent centuries. A great anecdotal evidence for the shift after 1945 in Western Germany can be found in the history of Tagesschau, Germany's most watched (and influential) news program. Being located in Hamburg, they originally used Sonnabend when referring to Saturday in weather forecast and alike - and received stiff opposition from southern Länder like Hesse, which resulted in a switch to Samstag in the late 1950s. Eventually the time when people started to care again for more than just rebuilding:)
Being the programming watched by almost everyone owning a TV, way into the 80s, their language formed national habit (*1). A bit like US Broadcast English or British RP.
Now what's a bit funny in that context is that not only the producing station (NDR) still uses Sonnabend in most of their other programming, but Tagesschau as well uses Sonnabend every now and then, most notably on their web pages.
Considering this, I'm pretty sure there will no over all 'win' in German speaking regions within the next centuries. It's quite solidified, much like Krapfen vs. Berliner vs. Pfannkuchen.
*1 - Their power in language forming has been shown just recently with the introduction of Belarus in all programming. Until then many Germans didn't even know that Weißrussland called itself Belarus - now, a bit more than a year later Belarus is thought to be the default naming.