First Officer reported excessive aileron trim was required to maintain controllability. The flight crew diverted to make a precautionary landing.

Narrative

Aircraft X was a red eye from ZZZ to ZZZ1 that departed at XA35pm on Date. Myself and the Captain arrived at the plane on time and proceeded to accomplish our duties that night. Everything was setup and double checked as appropriate followed by the corresponding checklist as well as the walk around. We pushed off the gate and continued to the runway without any issue. I made sure the flaps were set to 1 for takeoff. The rudder and aileron trims were also set and checked before takeoff. The Tower cleared us for takeoff; at which point controls were transferred to me since this was my leg to fly. The aircraft proceeded down the runway straight with little need for correction until rotation. Once the aircraft started to climb there was a gradual bank to the left that I began correcting with right aileron and small amount of rudder. It continued to get worse with the aircraft banking farther left as we climbed and speed increased. At first I thought it must have been very strong crosswind after takeoff but realized after looking at center display unit there was no wind. As we continued to climb I tried to adjust the bank with aileron inputs and trimming. I mentioned to Captain that the inputs are more than normal and to check to see if I was missing anything. The Captain then mentioned that it did feel a bit unusual in his seat with the aircraft feeling out of trim. At this time we decided to level off to further work the issue. The yoke at this time was indicating 7 on the aileron trim on the right side in order to accomplish straight and level. As speed started to increase; so did the need for more aileron. We immediately reduced the speed to 250 kts. to lessen the correction since we would be approaching the maximum input of right aileron if it continued. We transferred controls to see if it was something on the FO (First Officer) side malfunctioning. The Captain had the same issue and could not correct it. We then transferred controls back and began to look for an indication or something that could be the issue. There were no lights illuminated indicating a problem; so we started to look to see if we could find it on our own. We looked at flaps/slats; landing gear; trim settings; spoilers; speed brakes as well as all of the fluid levels like hydraulics. We could not find anything that was indicating the incorrect position or setting. The Captain then went through the QRH to find a checklist that would apply. Again nothing applied to the banking the plane wanted to do. We talked about diverting and what else we needed to do. The Captain contacted Dispatch; Maintenance; the fleet manager and Chief Pilot to see if they could help. They walked us through a few trimming ideas and had us double check system indications which did not fix the problem. We all then decided to [request priority handling] and head to ZZZ2. ZZZ2 was straight ahead and had long enough runways to land overweight. We descended at a slow rate and continued to Runway XXL. The Captain setup the FMS and corresponding instruments; we briefed the approach and completed all checklists. We transferred controls one last time so the Captain could land the aircraft. After putting in flaps; gear and slowing down the speed; we started to feel a slight rumble or vibration sensation in the controls. We were on speed and on the ILS completely stable the entire time. The Captain was able to land the aircraft safely even with the inputs still far to the right. We then taxied to the gate with emergency vehicles following us from behind.