I am interested in relocating my electrical service panel during a major renovation of my basement. Can I relocate the panel by running the service cable through the floor joists to the opposite wall from where it is currently located (approximately 75ft)?

I will assume the existing panel has the Main Breaker in it. If this is the case, once you move the panel, a new Main Breaker will need to be installed in place of the existing panel, or at or near the meter.

Yes, you can run a feeder cable through the joists.