**Solution:**

To find the equation of the locus of a point equidistant from two given points, we can use the concept of the perpendicular bisector.

Let the coordinates of the given points be A(1, 0) and B(0, 2).

**Step 1: Find the midpoint of AB.**

The midpoint of a line segment is the average of the coordinates of its endpoints.

The midpoint M can be found using the following formula:

M = ((x1 + x2)/2, (y1 + y2)/2)

In this case, the coordinates of A are (1, 0) and the coordinates of B are (0, 2).

So, the midpoint M is:

M = ((1 + 0)/2, (0 + 2)/2) = (1/2, 1)

**Step 2: Find the slope of AB.**

The slope of a line passing through two points (x1, y1) and (x2, y2) can be found using the following formula:

m = (y2 - y1)/(x2 - x1)

In this case, the coordinates of A are (1, 0) and the coordinates of B are (0, 2).

So, the slope of AB is:

m = (2 - 0)/(0 - 1) = -2

**Step 3: Find the negative reciprocal of the slope of AB.**

The negative reciprocal of a slope is obtained by taking the negative of the reciprocal of the slope.

In this case, the slope of AB is -2.

So, the negative reciprocal of the slope of AB is:

-1/(-2) = 1/2

**Step 4: Find the equation of the perpendicular bisector of AB.**

The perpendicular bisector of a line segment is a line that is perpendicular to the line segment and passes through its midpoint.

The equation of a line with slope m passing through point (x1, y1) can be found using the following point-slope form:

y - y1 = m(x - x1)

In this case, the midpoint M is (1/2, 1) and the negative reciprocal of the slope of AB is 1/2.

So, the equation of the perpendicular bisector of AB is:

y - 1 = (1/2)(x - 1/2)

Multiplying through by 2 to eliminate the fraction:

2y - 2 = x - 1/2

Rearranging the equation:

x - 2y + 3/2 = 0

Multiplying through by 2 to eliminate the fraction:

2x - 4y + 3 = 0

Therefore, the equation of the locus of a point equidistant from the points (1, 0) and (0, 2) is 2x - 4y + 3 = 0.

Hence, the correct answer is option A.