Given information:
- Acceleration of the elevator, a = g/4 (upward direction)
- Initial speed of the elevator, v0
- Distance of the ball above the floor at the time of release, h1
- Coefficient of restitution, e

To find:
Bounce height of the ball with respect to the elevator, h2

Solution:
1. Determine the time taken for the ball to reach the floor from the release point:
- The initial velocity of the ball relative to the elevator is the same as the elevator's velocity, which is v0.
- Since the ball is released from rest, its initial velocity relative to the elevator is 0.
- The acceleration of the ball relative to the elevator is the difference between the acceleration of the elevator and the acceleration due to gravity, which is (g/4 - g) = -3g/4 (downward direction).
- Using the equation of motion, s = ut + (1/2)at^2, where s is the distance, u is the initial velocity, a is the acceleration, and t is the time, we can solve for the time taken for the ball to reach the floor.

2. Determine the velocity of the ball just before impact with the floor:
- The initial velocity of the ball just before impact is the velocity of the elevator plus the velocity of the ball relative to the elevator.
- The velocity of the elevator just before impact can be found using the equation of motion, v = u + at, where v is the final velocity, u is the initial velocity, a is the acceleration, and t is the time taken.
- The velocity of the ball relative to the elevator just before impact is the acceleration of the ball relative to the elevator multiplied by the time taken.

3. Determine the velocity of the ball just after impact with the floor:
- The velocity of the ball just after impact is the negative of the velocity just before impact multiplied by the coefficient of restitution.
- Since the ball bounces back, the velocity after impact is in the opposite direction.

4. Determine the height reached by the ball after the bounce:
- The height reached by the ball after the bounce is the maximum height it reaches above the floor.
- Using the equation of motion, v^2 = u^2 + 2as, where v is the final velocity, u is the initial velocity, a is the acceleration, and s is the distance, we can solve for the height reached by the ball after the bounce.

5. Substitute the values and solve for h2:
- Substitute the known values into the equations derived in steps 1-4 and simplify to find the expression for h2.
- The expression for h2 will involve h1, v0, g, and e.
- Comparing the given options, we find that the correct answer is option 'B', which is e2h1.

Hence, the bounce height of the ball with respect to the elevator is e2h1.