Go to China Town first - you'll find that most of the musicals are right next door, so you can 'window shop' to see which grabs your fancy. Also in and around the West End are a lot of museums and art galleries, including the National Gallery, National Portrait Gallery and (heading north a bit) the British Museum. All of those are free entry, though may have special exhibitions which cost a bit - probably about £25 to £50 for you _and_ your mother to attend.
Similarly, there are other museums and galleries - the Tate Britain and Tate Modern, the Victoria and Albert Museum, the Natural History Museum and the Science Museum (the last three museums all in South Kensington).
If the weather is any good, then gather together a few snacks and head to one of the parks for a picnic - Hyde Park/Kensington Gardens, Regents Park, Greenwich or a few others. If your mother is young at heart, head off to Camden (one of the UK's biggest tourist attractions!)