Instead of a timer you could have a counter. The Brita filter looks to last 40 gallons. My Brita pitcher holds 10 cups. 10 cups equals 0.625 gallons. Which means my filter lasts for ~64 fills of the pitcher.

So using a counter until 64 would tell me to refill. All you need is a plus sign and some way to reset.

Now that I've done the math I might start writing this on a pad next to my refrigerator because I never have any idea when to switch the filter. Paper is preferable to electronics in my life.