I can't seem to ever predict where D.VA is going to go when she ejects from a dying Mechasuit. Is it player controlled, or is there a predetermined direction that she can go?
She ejects out the back of it, so she's left standing directly behind where the mech died. The reason you can't predict where she's going to go is that as soon as she's ejected out the back, she has a moment of invulnerability where she is free to move and dodge attacks. This is to prevent players from immediately one shotting her as soon as she pops out the back of the mech.
Steer D.va's Ejection
When your mech gets destroyed, you can hold down the direction you want D.va to eject out at and she will jump out that direction. This means you can hold your strafe, forward, and backward keys down while getting ejected and she will pop out that direction. By default, if you don't press any movement keys, D.va will eject backwards away from her mech.
I play a lot of D.Va and can confirm that I try to eject in the direction that will put the mech in between me and enemies and/or get me closest to cover.