As you travel close to the speed of light, it is to my understanding you gain mass. Does this also apply when the brain sends electrical signals to the muscles? Do the signals (that are traveling at the speed of light) cause the body to weigh more?
John Rennie's answer is correct.
Even if the signals would travel through or nerves at relativistic velocity, transmitted by some particles, this would not increase our mass, because to give those particles the energy to reach that speed, we have to consume it from somewhere else (perhaps to burn some calories). So the mass they gain by moving near the speed of light is in fact given to them by your body, and there is no increase of mass.
Let's discuss instead of a human with signals traveling through nerves, about a robot, which has optical fiber wires in which information travels at the speed of light. To create photons to be transmitted through the fibers, the robot consumes some energy. This energy is (partially) preserved in the photon. So if the photon increase the robot's mass due to its own movement mass, it is the mass which was "burned" to emit the photon. So its mass doesn't increases because of the photons traveling through its optical fibers wires. I can say that it loses mass, because these processes release heat (energy) in the universe. But this variation is too small anyway.
The signals travelling from your brain along your nerves travel at much much less than the speed of light. The maximum speed is about 100 m/sec.
In any case what travels down the nerve is a change in the sodium and potassium ion concentration not some mass, so there isn't anything to gain relativistic mass.