When attempting to assign a newly-created budget to a project from the Budgets screen, the project cannot be selected on the grid. This might be because the project already has a budget assigned to it.
To resolve this:
From the main menu, open the Projects screen.
Select the project on the grid to which you want to assign the newly-created budget and view its details.
Under General > Billing Options tab, you can find the budget assigned to the project under the Fee Schedules, Budgets, and Estimates section. Select the new budget from the drop-down and click Save & Done when finished.