Oooh...
Since I'm from Lower Saxony, especially where Henry the Lion reigned and died. He's the first duke of our city of the Welf dynasty that ruled us till the end of the German Empire, he's also the one who developed our city very much and even gave us our cathedral and more important our symbol, his lion as our city symbol and as a statue in the center. So I'm really happy to talk about the duchy of Saxony and Henry. So, Saxony was a big and powerfull stem duchy with a good position for it's time and for the future.
If you want to go with Henry, then either give him more luck againts his older cousin, the Emperor Frederick I (Barbarossa), so first make Barbarossa give Henry the imperial city of Goslar, something Barbarossa often refused, so let's just say that he finally gives in. After that make Barbarossa more successful in Lombardy, because he lost there he blamed Henry for not being good enough to help, thus he invaded Saxony and forced Henry to go to exilled in Normandy for 3 years. So let's make Barbarossa win and praise Henry, shortly before we kill him similar how he died on the crusade. Thus him dying earlier makes his now 11 year old son Henry VI. Now with nobody to really stand in his way neither can Henry VI break up Saxony nor will Bardowick revolt againts Henry, so the city can become bigger.
Now you could either make Saxony and Bavaria slowly unite, thus building an OP Welf base in the Empire, thus we get a much more centralized HRE under the leadership if the Wefs, if this Welfish HRE survives then we can expect Low German to have a bigger influence in German/Dutch or it even replacing High German as the official language of the "German" part (German being Non-Romance areas). We could even expect the Low Lands, Northern East Germany and even maybe Jutland becoming core regions of a bigger Saxony, thus being nearer to the political and economic center of the empire, plus the much more similar linguistic and cultural relations with Low German (or rather Saxon) the Low Lands will probably not leave, so you can expect German colonies, where they start is up to you.
Now if you want a Strong Saxony that isn't integrated into Germany, then just have have them do a bit worse after Henry dies than the scenario above. Let's say that they loose Bavaria and everything below Saxony, they also don't become Emperor because of wide dislike of the Saxons, they become to weak to take controll themself but also are strong enough so that no sane emperor would invade Saxony. They pretty much become a somewhat pariah that is still economical and politically bound to the empire. So they will still probably integrate the regions east of them above the benrath line (which is the border between High and Low German). So Mecklenburg, Brandenburg and depending how the Empire will expand, but considering that Hither Pomerania was added to the Empire shortly after the 1170s, expect the Saxons to also expand to OTL Imperial border to Poland.
To the West I can see Saxony expanding to Frisia and maybe a bit more to the Rhine, maybe till the Sieg if my geography isn't failing. Maybe, but that's just a maybe Saxony can expand to the Netherlands north of the Rhine, that would make Saxony in the long term even stronger and more of a colonial and sea power. Up north maybe they can conquer Jutland from the Danes if the get the chance. So we have Saxony being the dominant power in North Germany, being the only way for the Empire to get to the Baltic and North Sea, with enough power to survive as it's own body and political and cultural differences to the rest of Germany. I wouldn't be surpised if they left the empire around maybe the 1400s if the Empire isn't offering them anything. After that they can make the North and Baltic Sea their Mare Nostrum, influencing Scandinavia even more and giving England much more influence than Dutch and German (both Low and High) ever did OTL. Hell, Saxony could even make it to either conquer or integrate England through a personal union, with a shorter Norman and Parisian French influence English would become much more Germanic than it became OTL, maybe it would stay like Early Middle English in a lot of ways, maybe even loosing the TH sound like it's continental brothers. After the 1500s and 1600s Saxony could become a colonial power, colonising North America. It would probably become a Rival to France, the Empire, Scandinavia (either a united Kalmar or only Danish (maybe Swedish, but I could see them rather as allies)) and Poland. If the Teutonic Order survives I could see it becoming a Saxon puppet so to be save from Poland, Lithuania and Denmark.
That of course if there isn't a personal union between Saxony and the Scandinavians. Maybe they will form a Northwest-Germanic creole language. (kinda like how it was after East Germanic split from Proto-Germanic)
Low German would also be a bit different than OTL with less Dutch/High German influence, rather I would see Low German influencing Dutch, maybe even assimilating it. With German I would see a lot more borowings for maritime, colonial and foreign (like America, India, Africa and East Asia) concepts.
Uuuh the possibilities are endless...
Suffice to say, the ideas of a (Low) Saxon super power gives me quite the hard boner, like wurtzite boron nitride kind of hard. So thanks for the post OP.
PS; I even made a late HRE CoA if the Welfs became Emperors (I used Henrys the Lions CoA, as I thought that he would become to prominent that the Welfs won't change their CoA)