A B737-300 flight crew suffered an altitude excursion when a modified descent clearance was incorrectly programmed into the single FMC. The failure of the PF to verify the reprogramming was cited as contributory.

Narrative

We were cleared to 'descend via' a STAR into our destination. We were flying a -300 in VNAV. ATC requested we delete the restrictions at some waypoints and keep the restrictions at another. The First Officer (PM) actually made the changes to the FMC programming and it happened quickly. As the PF; I should have made the FMC changes myself; but with a single CDU -300 model the First Officer jumped in and reprogrammed it. I failed to verify the new programming and we left 8000 for 7000 early. ATC requested we climb back to 8000. We quickly climbed and the rest of the arrival and approach were completed without incident. On reflection; neither the First Officer nor I are certain exactly how we left the altitude early. It could have been a programming/verification error or we could have slipped into V/S and drifted down. I know we disconnected the autopilot and climbed quickly back to 8000 when requested by ATC. A small factor in my failure to effectively monitor was the distraction caused by a failed RADAR altimeter which Maintenance addressed on arrival. Programming the FMC multiple times and sending a maintenance request late in the arrival; led us into the event.I am opposed to VNAV arrivals with 'descend via' clearances in aircraft with only one FMC/CDU. The VNAV arrivals with a -700 model seem to make maintaining situational awareness much easier. No ACARS maintenance requests below FL180 might be smart. There is no substitute for verification of FMC programming.