You can file a flight plan in ForeFlight using a call sign instead of your aircraft's tail number using one of two methods. Method 1 is better if you only file with a call sign on some flights, or file using different call signs, and Method 2 is better if you always file with the same call sign for a given aircraft.
Method 1 uses the optional Call Sign field that appears beneath the Aircraft field in the flight plan filing form. Here is a video that describes how to do this:
Video 1: How to file with a call sign.
To file with your call sign, enter your call sign using the contraction and number in the Call Sign field. If your flight plan is filed using the domestic form, your tail number will be added to Remarks. If your flight plan is filed using the ICAO form, your tail number will be placed in field 18 after the REG/ label (registration number), for example REG/N01234. If you don’t enter a call sign, the flight will be filed under your N-number.
Image 1. Enter your call sign in the Call Sign field of the flight plan form.
Method 2 is a more permanent solution that uses the REG field in the Other Information section of your aircraft's profile.
Go to More > Aircraft and tap on the aircraft you want to edit. Tap on the Tail Number field at the top of the profile and enter the call sign you want to use when filing with that aircraft. Then scroll down the page and tap on Other Information, under the Filing header. Tap on REG in the Other Information popup and enter your aircraft's registered tail number.
Image 2. Enter your call sign in the aircraft's Tail Number field and enter the tail number in the REG field under Other Information.
Now when you file a flight plan using that aircraft, the call sign that you entered will be used as the tail number, and the tail number you entered in Other Information will be placed in field 18 of the ICAO form after the REG/ label, just like in Method 1.
Checking Your Flight Plan
You can check what your flight plan will actually look like when it gets filed using the ICAO form by tapping the Send To button at the bottom of the Flights view filing form (visible near the center-bottom of Image 1) and tapping "Export PDF". This will generate a PDF of your flight plan in the official ICAO form.
If you used either of the methods outlined above, you should see your call sign in Field 7 "Aircraft Identification", and your aircraft's tail number in Field 18 "Other Information" after REG/.
Image 3. The ICAO form PDF should show your call sign in Field 7 and your aircraft's tail number in Field 18, after REG/.