I must heartily agree with choosing your own site symbol. If you're starting a fortress, choose its name, and choose the name of the group going, you should be able to choose the symbols that this group will rally under, perhaps from a list given for each Civ - you know, to avoid some poor group's symbol being the aforementioned Elf. Or a unicorn. Or the purring maggots I got for my first fortress. I mean, sure, the maggots are useful (kind of?), but they're still vermin. How many of my Dwarves detested purring maggots? About half. I'm surprised they didn't get unhappy thoughts just walking through my carved hallways.
And yes, any language-based thing you choose should also have a random option. This has, after all, been the precedent. You can name the world or use a random one, name your fortress & group or use a random one, I would think the same would go for the symbol. Of course, you only see the symbol at all if you have a habit of looking at your engravings and sewn images, but for those of us that do it would be a nice touch.
I'd also like to see the Civ symbol, too, just for when I get the cool-sounding ones, but I will also admit that it's much less important to me to be able to control that symbol.
... At least until my fortress becomes the capitol.