Introduction:
In order to determine the day of the week for October 3rd, 1437, we can apply a mathematical algorithm known as the Zeller's Congruence. This algorithm allows us to calculate the day of the week for any given date in history.

Applying Zeller's Congruence:
To determine the day of the week for October 3rd, 1437, we need to calculate three variables: A, B, and C.

Step 1: Calculate A
The year 1437 falls within the 15th century, so we need to subtract 1 from the year and divide it by 100: A = (1437 - 1) / 100 = 14.

Step 2: Calculate B
The year 1437 falls within the 15th century, so we take the last two digits of the year: B = 37.

Step 3: Calculate C
The month of October is considered the 10th month, so we subtract 2 from the month value and assign it to C: C = 10 - 2 = 8.

Step 4: Applying the Formula
Now that we have the values of A, B, and C, we can utilize the Zeller's Congruence formula:
h = (q + [(13(m+1))/5] + K + [K/4] + [J/4] - 2J) mod 7

Where:
- h: the day of the week (0 = Saturday, 1 = Sunday, 2 = Monday, ..., 6 = Friday)
- q: the day of the month (3 in this case)
- m: the adjusted month value (C in this case)
- K: the year within the century (B in this case)
- J: the zero-based century (A in this case)

Step 5: Calculating the Day
Plugging in the values, we get:
h = (3 + [(13(8+1))/5] + 37 + [37/4] + [14/4] - 2*14) mod 7

Simplifying further:
h = (3 + [117/5] + 37 + 9 + 3 - 28) mod 7
h = (3 + 23 + 37 + 9 + 3 - 28) mod 7
h = (87) mod 7
h = 3

Conclusion:
According to Zeller's Congruence, the day of the week for October 3rd, 1437 is 3, which corresponds to Tuesday. Therefore, the correct answer is C) Tuesday.