Once returning home from a day on the ice immediately removing any dead minnows from the bucket and placing an aerator into the bucket will give your minnows a fighting chance at staying alive until your next voyage onto the ice. When dead minnows remain in your bucket, they continue to break down, which causes the water quality to decline which in turn put more stress on the remaining minnows and will kill the weaker ones. A simple pet store aerator pump and a few of the larger air stones are all that you need to keep you bucket filled with sufficient oxygen for your minnows.
Another key to keeping minnows alive in the winter between fishing trips is to change the water every couple days using non tap water. Sump pit water works well, as does well water. Regular tap water may be used, just be sure to add a de-chlorinating solution that you would use for a fish tank. Once you have added the proper amount of solution, you must then set the water outside to allow to cool to the temperature of the water that the minnows are currently occupying.
There are also several products available that will help keep your minnows lively. Bait Saver products balance the water your minnows are in, lower minnow stress, provide the minnows with an extra protective slime coat, any may even add electrolytes which are known to keep minnows lively.