The average dwarf should prefer doing traditional things . . . i.e., for food, "Plump helmet roasts were good enough for my pappy and good enough for my grandpappy, so dammit they'll be good enough for me! Don't want none of this stinkin' muskox biscuit!"
This could manifest as each fortress having a distinctive profile of foods that are most often eaten, symbols most often seen, materials and styles of clothing and decoration, et cetera. Dwarves would react unfavorably at first to new things, but with time and player interference, the profile could shift: after a while of only having rope-reed kilts to wear and eating only fish, the dwarves of the fortress would start prefering rope-reed and kilts and fish to the traditional mountainhome garb and chow.
Where dwarves have choices in terms of what they make (say instead of making specific clothing, a tailor is told to "make clothes"), their products may be weighted toward the fortress' cultural norms.
The profile could then factor into reputation: wealth might attract immigrants, but immigrants might also be attracted by things about the culture profile that they like and dissuaded by things they don't. A cosmopolitan fortress with many choices as part of its profile might be viewed as a favorable place to settle down by anyone, while an insular fortress with one industry (say a desert fortress specializing in hunting camels for food, clothing, and crafts) might turn out to be small, inbred, and hostile toward outsiders.
Traditions could encompass particular types of jobs (Urist McFisher is proud to be a fisherdwarf like his father) as well as products, materials, and styles.