Let Juanita's cousin's age be represented by \( x \). Since Juanita is 13 years older, her age can be represented as \( x + 13 \).
According to the problem, the sum of their ages is no less than 103 years:
\[ x + (x + 13) \geq 103 \]
This simplifies to:
\[ 2x + 13 \geq 103 \]
Next, subtract 13 from both sides:
\[ 2x \geq 90 \]
Now, divide both sides by 2:
\[ x \geq 45 \]
Thus, the youngest age Juanita's cousin can be is \( \boxed{45} \).