To determine the value of $$b$$ such that the work done on the particle is zero, follow these steps:

The work done by a force on a displacement is given by the dot product:

$$\text{Work} = \overrightarrow{F} \cdot \overrightarrow{d}.$$

Given the force

$$\overrightarrow{F} = 2\hat{i} + b\hat{j} + \hat{k},$$

and the displacement

$$\overrightarrow{d} = \hat{i} - 2\hat{j} - \hat{k},$$

compute the dot product:

$$\overrightarrow{F} \cdot \overrightarrow{d} = (2)(1) + (b)(-2) + (1)(-1).$$

Simplify the expression:

$$\overrightarrow{F} \cdot \overrightarrow{d} = 2 - 2b - 1 = 1 - 2b.$$

Since the work done is zero:

$$1 - 2b = 0.$$

Solve for $$b$$:

$$2b = 1,$$

$$b = \frac{1}{2}.$$

Thus, the value of $$b$$ is $$\frac{1}{2}$$, which corresponds to Option B.