How should you handle a team member that has become rather complacent but is still a key member of the team?

Give the teammember a challenge, which will take a lot of his time and which will get the best out of him/her. Next you will need to learn the teammeber that being complacent is not always good. Through a good conversation and communcation the team could get a better functioning.

Dom you mean conceited??? 

Challege the team member with a more meaningful assignment or task.   Routine makes anyone complacent and tends to direct him to be satisfied with what he is regularly doing.   Keep him on his toes and always bring out the best in him by motivation thru delegation of very important task showing him that he has your confidence  to complete the assignment.

Try to let the member learn him or  herself by providing an opportunity learn being complacent is not always good, you could let him or her handle some important matter in the team to learn this.

Plenty of support and encouragement. Positive reinforcement. Add a more challenging work strategy. Make him/her leader. Assign role of importance to him/her.

This kind of people didn't wake up in tn the morning and find him like that.

NO, it started a long time ago but, you as a superior didn't notice it, and also you might be a main reason for that.

All you have to do is to make him feel safe by conversation to discover what are the real reasons

to get him or her to do your bidding, you have to give him or her a more dearing task that demands most of his or her time to perform.

engage the person in meaningful conversation. effective communication is very essential in a project team