4

If you say the aim of Ben, you can also say Ben's aim.

If you say the objective of the project (which is not a person but an object), do you say the project's objective or the projects objective?

3

project's objective

The objective "belongs" to the project, so you would use 's to indicate ownership. You can say "the project's name" or "the project's staff".

1

You could certainly use the apostrophe-s possessive about a project, but it would be unusual. We don't tend to use this this form when a thing is the possessor (see this answer).

You could stick to the of-possessive form, but for project we can also make a compound noun, for example project objective, project name and project team.

There is a lot of variation in which form is preferred: see these NGrams for project objective, project name, project staff and project team

Similarly, with hill, you can say the top of the hill or the hilltop, but hardly ever the hill's top;

Not the answer you're looking for? Browse other questions tagged or ask your own question.