Answer
Tension should be multiplied by a factor of four.
Work Step by Step
The formula for the speed of a wave on a string is $$v=\sqrt{\frac{F_T}{\mu}}$$ If the new tension $F'_T=4F_T$, the new velocity is $$v'=\sqrt{\frac{4F_T}{\mu}}=2\sqrt{\frac{F_T}{\mu}}=2v$$ Therefore, the tension must be multiplied by a factor of four.
