What is a good “test project” to prove one’s skills as a UI designer (to work with a UX person) for a website development company, e.g, if one’s portfolio is out-of-date, but they have prerequisite talent & skills?

Let’s frame your question in the following:

You’re applying for a web development company
Your portfolio currently would not demonstrate your superpowers
But a Test Project will
You are applying for a User Interface Designer position
In the company, they have a UX Researcher and a UX Designer in the team
You would like to foresee the Test Project that they will give you to be able to prepare for it

It Depends on the Role and Responsibilities

Now, it depends on the structure of the company’s Design Team. In some, the UI Designer is focused on the following:

Interpreting the Product Requirement Document (PRD) and the Site Map & Task Flow to come up with
- Navigation Design
- User Interface Design in the form of Wireframes and Interaction Diagram
- Presenting and Testing these Artifacts thru Prototyping

In some Design Team structure, the UI Designer is also the Visual Designer – who is responsible for the Visual Design aspect.

And in some, the UI Designer is also the Interaction Designer and Visual Designer.

And in some, the UI Designer is THE Designer.

Whatever project that may be given to you, just remember to take care of the following:

Each step of your process must demonstrate collaboration and validation (ask questions and get clear answers)
Take care of the Visuals as well as the Interaction (be able to rationalize the whole picture rather than how awesome the gradient is)

So the test might not be 100% about the destination but the quality of your journey itself.