A B737-300 'B' autopilot overshot an altitude on climb out and the crew received a TCAS RA. Maintenance MEL'ed the 'B' side but on subsequent flights the 'A' autopilot also failed to capture the selected altitude.

Narrative

In the climbout we were cleared to 8;000 FT. Traffic was reported at one o'clock; five miles; CL-30 at 9;000 FT. The 'B' autopilot was engaged at approximately 3;000 FT MSL; and then the Level Change mode was engaged. As the aircraft approached 8;000 FT MSL; I was scanning outside our aircraft for the CL-30. The First Officer reported that the altitude did not capture; the Level Change mode dropped out; and the Stab Out of Trim light illuminated. The First Officer arrested the climb at 8;800 FT MSL and began to correct to the assigned altitude. Passing 8;600 FT MSL; ATC called 'Altitude Alert.' We had an accompanying RA to descend; which we were in the process of doing. This aircraft had a previous writeup for the 'B' Autopilot concerning altitude deviation/capture problems of approximately 100 FT to 200 FT; to my recollection. Tests were performed and aircraft was returned to service with nineteen legs being flown prior to this event. Upon taxi in and shut down at our destination the 'B' autopilot was deferred by Maintenance. As we monitored the aircraft in subsequent level offs (two times to my recollection); it continued to demonstrate the autopilot errors approaching 200 FT before correcting. At this point; the autopilot use was discontinued for the duration of the flight.