I think the point of billing by day (or week) is because the project is unclear. They can change the spec as much as they want, but they're paying you for all your time changing it.
The time estimation is for fixed bid contracts, so you can figure out what your effective rate is and how much time you can't devote to other clients. It's not for them to put a cap, e.g. $1000 a day up to 30 days, so we're capping at $30k. If you completed in 25 days, they would pay $5k less. If you complete in 35 days, you only get paid for 30. There's no upside in that for you, only for them.
A friend of mine handled this by giving a rough estimate of how many days he thought it was going to take and then giving them the option of a fixed price contract, e.g. "I think it's going to be X hours at $Y/hour, though it could run longer. If you want this for a fixed price, it's going to be 2.5 times X times Y, no more, no less". It's then up to the client to make the risk assessment of high fixed cost, or open ended cost.
But really it just comes down to your estimation, and if it's not clear what to estimate it by, then you should probably be billing for unit of time rather than fixed.
