After reaching 100 conjuration, I was excited to head over to the Mage College and start the Sigil Stone/Unbound Dremora side quest.
I reach the top of the roof, and manage to summon the Dremora, but as soon as I summon it a Dragon decided to visit.
I finish off both the dragon and Dremora with rather ease, but I cannot summon the Dremora again. No matter how many times I try the Dremora will not appear so that I can tell it to grab me the sigil stone.

Reload an earlier save and try again. Perhap try find and kill the dragon so it can't interfere.

These were actually both of the first things I tried, but to no avail. When I cast the spell, I hear a sound but no Dremora appears.

Things I have tried:
Restarting from a previous save.
Waiting in-game for about 2 days and spoke with the quest giver again.
Looking around for the Dremora to see if he summoned elsewhere.
Restarting the client.

Is there some way to clear the game's memory cache, or even a console command to summon this Dremora?

To clear the memory, just quit to desktop. Come back and then reload the earlier save.

The game seems to keep some stuff in memory from just reloads.

Well, I have restarted the client numerous times. I even re-installed the game.
Something within the code got botched up when the dragon flew into the college.

I can hear the spell make the resolving "clink" sound, but no Dremora appears.

Really, I think the only solution is either wait for a patch, or use the console commands to get a Dremora out.

I figured it out!
This is what worked for me.

I opened the console and typed

player.placeatme 00099F2F

then I typed

player.placeatme FF001046

Now both of these commands summon a "Unbound Dremora", but I am unsure which one is the one related to the quest.
Just kill them both, and the code resets or something (who knows) and I was able to summon the Dremora normally and continue with the quest.

I hope this helps anyone who has an issue with this in the future! (This console abuse may work out more bugs as well if applied correctly)