Eeryone will have their own idea, but for me it's got to be Al Haramein. An old Ottoman house, simply restored, down a quiet lane off a busy street, 10 minutes walk away from the old city.
Nice staff, and all the taxi drivers know it too.
They have shared rooms [clean sheets] and, in summer, sleeping on the roof [matresses supplied] is even cheaper. Showers in the basement are spotless and hot water all day.
Downside - don't bother having breakfast there - it;s ok-ish, but there's a nicer little kiosk-cum-cafe a bit further up the same lane.
Can't give you current prices but they have a website and I paid about 200 Syrian Pounds a couple of year ago in a shared room.
If you want private, en-suite, but still cheap, Al Rabie is a couple of doors away. Owned by the same family.