Old Way - You throw out a complete guesstimate of $50,000 (500 hours). You are now a budget scapegoat.
New Way - You think it will be about 500 hours, but you present it with this wording:
Boy, this is really hard to say. I apologize for the range, but
somewhere between 500 - 1250 hours. I can 100% guarantee that 1250
hours is plenty of time, but there is only a 10% chance that the
project will come in on budget with a shoestring budget of 500 hours.
How many hours would YOU like to dedicate to this project.
Being a responsible estimator is important, but being responsible for your estimates is not. You have to build the solution, you should not also have to manage the budget. Put your estimations in ranges with probabilities and let your manager manage the costs and risks of the project.