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One way to utilize a reminder unconventionally within a For loop is to use it as a conditional statement. For example, instead of iterating over a sequence of numbers, the loop can iterate over a sequence of characters and use the reminder to check if the index is even or odd.

Here's an example in Python:

word = "hello"
for i in range(len(word)):
    if i % 2 == 0:
        print(word[i].upper())
    else:
        print(word[i])

This code will print out the word "HeLlO" by iterating over each character in the word and checking if the index is even (using the reminder operator) to determine whether to print the character as uppercase or lowercase.