Before I answer that, let’s first talk about the traditional business analyst role. Typically this is a person that gathers and/or decomposes business requirements into software requirements. So this is the person that, on paper, straddles the technical and business sides of the house.
When we look at Scrum by the book, we have a product owner, a team, and a scrum master
The business analyst could potentially act as an assistant to the product owner, because product owners typically have more responsibility than time. The business analyst could help with the backlog grooming, the prioritization, the definition of done, etc. BE CAREFUL! Do *not* make the business analyst a “proxy” between the team and the product owner. In other words, the product owner still needs to talk to the team as much as possible, but can also leverage the “extra hands” of the business analyst.
Another option is that the Business Analyst becomes a scrum master, especially if this person has the facilitative/servant-leader mindset
Some business analysts have a lot of experience working in a consultative/facilitative role. Making them the Scrum Master continues to allow them to straddle the gap between the team and the business. Instead of translating business requirements into technical requirements, they will be responsible for facilitating conversations, removing impediments, and improving team performance.
Lastly, if the business analyst has a technical and/or ui/ux background, they could become part of the team
The Scrum framework doesn’t out team member types such as business analyst, dba, front end engineer, back end engineer, it simply calls out the team. Another important thing to remember is that cross functional teams are better suited for Scrum and also decrease the risk (truck factor) of your team. In a perfect world all team members would be generalists. In reality the former business analyst could pair with and cross train with other team members and be able to contribute his or her unique skills and pick some new ones up along the way.