Can I create a button to add a new column to my base table?

Each row of my table is a problem statement.

There are columns which contain potential solutions that the team can vote on for their favorite solution.

I would like to give team member the ability to create new columns so that they can add new solutions on demand.

Is there a way to do this in the interface designer?

No. Typically, that data structure would be served by adding new rows as opposed to new columns. that may take the form of one table for Problem Statements, linked to another table of Potential Solutions where each row is a solution.