Does it have dual fuel tanks, If so, it may be the dual fuel tank valve assembly. The fuel pump should not be involved with the problem. There could be a loose connection from 12 V to the gauge, the wire to the fuel level sender could be loose on the gauge, have a break in it, have a loose connection to the sender. The sender could be bad, or the ground on the sender could be bad.

Back-track a little. You should have also replaced the *fuel float inside the gas tank.

fuel pump wouldn't do that, It does sound like the fuel float/sender unit, maybe first check your dad's rewiring, could be a bad connection there

Probably the float is going bad. if you go so far as to replace the float, best replace the pump as well. Another thing is, if he did some rewireing, there may be a short. That would also make your guage move.

GM cars fuel guages tend to do this, I have a UK vauxhall that looks like a buick or a German Opel and its fuel gauge bounces like that

poor ground wire connection at the sending unit

The clue is in "your Dad had to rewire it".

that would be your gas tank sender unit.

In my 1999 chevy blazer ls, the gas gauge has recently been jumping back and forth from E to F. The fuel pump has been recently changed. My dad saI'd he had to re-wire it. I don't know what it could be. Can it be a relay? Or should I just replace the whole float mechanism? Or the whole fuel pump? Need specifics please.