Idea #1: Community Picnic!
Plan a picnic with the family and their neighbors at a park or other grassy space within walking distance of their home. Either bring enough culturally applicable food for everyone or ask each family to bring something to contribute. Spend time talking, playing games, doing outside activities with the kids, and just hanging out! This is great because it brings you as the mentor team closer to the family AND their community. The family can also use this time to make better relationships with their neighbors! If you have done this before, please add specifics of what you actually did!
Idea #2: Meal with the Family
Make and share a meal with your refugee family! Go to their house and make something American...although they may not like it. Or you can try to make something similar to their typical cuisine. Make sure to double check about allergies and cultural/religious taboos.
Idea #3: Board Game Night
Have some fun and simple board games or card games? Teach them to the family! UNO is not difficult to teach even when there is a language barrier. Make it an opportunity to teach some simple words that are used in the game (colors, numbers, directions, etc.).
Idea #4: Picture Albums
Bring a few of your picture albums of your family, vacations you have been on, and good memories. Show the pictures to your refugee family and try your best to explain interesting things. Let them learn about you! If they have any photos, the family will almost certainly bring out theirs!
Idea #5: Movie Night
Bring a fun and age appropriate movie to their house (if they have a TV and DVD player) and watch it with them with the subtitles on. If they do not have a TV, maybe bring your laptop or invite them to your house!
Idea #6: Talk About the Flu
Use the documents here: http://www.cdc.gov/immigrantrefugeehealth/resources/index.html They are seasonal Flu Materials for refugees--several documents on the flu and how to prevent it in a dozen languages.