It is fun to go downtown Memphis (during the day of course) and riding the trolley. It's cheap, on weekends it's like $.50 I think and you want to take the Riverside (River..something) trolley. This takes you around town & down by the Mississippi River side & it's fun to do. Walking around downtown is fun too. You have to go to the Peabody Hotel & see the Peabody Ducks. They are so cute. They are usually out I believer from 11am-5pm. If you can get there when they come or leave the fountain, it's quite a little show.
You can also drive to the River, park & picnic there (course right now that may be a bit chilly).
You can look at parks to go to for picnics and play. Google nature trails & see what's in the area. We've not taken advantage of those yet.
Also the Memphis Zoo has a yearly pass that, if you go more than say 3 times it pays for itself. Then you can go as often as you like & don't even have to pay for parking (which I think is like $3).
We've been here 2 years but it's been kind of a frantic 2 years so we haven't availed ourselves to too much here either. We have a daughter getting married in January so we keep hoping things will calm down after that...although I'm sure it'll just shift gears.
You can also contact the Memphis Chamber of Commerce and they may have lots of ideas for your family that are within your budget.
Have fun!