I remember a time when we did keep Busy Lizzies in the house. With improved propagation and plugs they are usually replaced in spring. Used to see them with a few flowers at the end of a long tired stalk with no leaves.

Take a few cuttings and pop them in a glass of water - as late on in the year as you can.....they should root up and keep going over winter there - and can then be potted on early next year. However - like Dorjac says - it is easier nowadays to start afresh each season - unless it is a very special one....in which case - keep the stock plant AND do cuttings...belt and braces trick...LOL