Air carrier Captain reported a ground conflict on the takeoff roll with an aircraft landing on a crossing runway. The landing aircraft missed the ATC assigned turnoff; resulting in the departing aircraft to reject the takeoff to avoid a conflict on the runway.

Narrative

While we were taxiing to takeoff on runway 33 at DCA we ran the before takeoff checklist; and as soon as we got to runway 33 we were told to line up and wait. I called for the below the line portion of the checklist; as we were just finishing up checking runway heading we were issued our takeoff clearance. I advanced the thrust levers and started the takeoff roll. There was an aircraft that had landed on runway 1 and was advised to take November for the tun off. They missed their turn on November and rolled down and turned on runway 33 instead. Once I and DC tower noticed what happened tower told us to cancel out takeoff clearance. I pulled the thrust to idle and hit the breaks. We were at a max of 20kts when we rejected. Tower the advised us to taxi down to runway 22 turn left and another left on hotel and into the hold bay for 33 and to let them know if we need any assistance. We complied with the instructions and advised them we did not need assistance. Once I got the break set I called the FA's and let them know what was going on.. They made an announcement to the pax. After talking to them we ran the rejected QRC/QRH. I then called dispatch and Maintenance Control as we were waiting for out 15min Brake Temperature Monitoring System (BTMS) cooling time. After everything was done we resent for takeoff numbers and reran the taxi and before takeoff checklist. Then I called the FA's again to just double check they were still secure and everyone was good to go in the back. We then advised ATC we were ready. The rest of the flight continued as normal. Landing traffic didn't follow instructions and turned on a active runway. Landing a/c need to be aware of what is happening on crossing runways.