Pretty much the only family friendly option for the area, the Samurai Museum has an impressive display of costumes, swords and all things samurai to enjoy. The museum is relatively small, but staff are knowledgeable and always willing to answer questions from visitors in English. They have demonstrations by trained actors and you can try on different costumes too. Guided tours are available and a good option, so you can have in-depth explanations to all the sections.
Access: JR Shinjuku Station East Exit (8 minute walk) or Seibu-Shinjuku Station (4 minute walk)