Question

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

Answer 1

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. 

Answer 2

The speed would be reduced to half of the original speed.

Explanation

A rocket is propelled through space due to the forward momentum gained during thrusting for the amount of time the rockets were fired.

The amount of rocket firing time is proportional to the forward momentum obtained by the rocket to go through space.

If the mass of the astronaut is doubled but the amount of time the rockets were fired were kept same, this means that the momentum acquired by the rocket with doubled mass astronaut is tend to be same with the momentum acquired by the rocket without any change in the mass of astronaut.

It is known that momentum is directly proportional to mass and speed. So if the momentum tends to remain constant and the mass has to be doubled, then in order to propel with the same momentum, the speed should be reduced by half.