SSZ

my brothers 04 Envoy 5.3L. about a month ago the starter took a dump. it would only do a single click but wouldnt turn over. i put in a new starter and it was back to normal for about a week. then the single click would happen again once in a while. he also said that sometimes the starter would stay engaged for 2-3 seconds after the vehicle had started. i was thinking there could be an issue with the ignition switch?? i have tested the battery and it is strong. one last thing is that sometimes he has to try twice to get the vehicle started, what i mean by that is he turns the motor over once and then has to turn it over again for it to start. Any suggestions as to the problem?