A rocket is moving through space on one thruster, if the mass of the astronaut is somehow doubled, but not the amount of time the rockets were fired, then the speed would be

While the rockets are being fired the force is acting onto astronaut. According to a Newton's second law the formula is:

F = m * a

After the mass of the astronaut is doubled the force remains same. In order to get everything correct acceleration must be halved.

F = 2m * a/2

Smaller acceleration would result in smaller final speed.