What is the opportunity cost of doing something in your spare time? My university is using this textbook. The text clearly states, “Economists use the term opportunity cost to indicate what must be given up to obtain something that is desired.”

This leads me to believe that if you are a salaried worker who makes 50 dollars per hour and works a standard five-day workweek, the opportunity cost of you mowing your lawn during the weekend is 0 dollars. You are giving up your unpaid free time to mow the lawn. When I brought this up in the online discussion I was told by the instructor, "the cost of your free time is estimated using your wage rate."

Is free time = wage rate just the standard convention? This hardly seems appropriate since you aren't giving anything up other than your free time. The situation would be different if you were giving up weekend work to mow your lawn.
There is a crucial detail missing in the scenario you describe.  Is the worker working a five day week (and no more) because that is what their terms of employment require?  Or do they have the option of doing additional paid work (at the same rate, and for as many hours as they wish) at weekends, but on this occasion have opted to work only five days in the week?  If the latter, then it is clear that the opportunity cost of mowing the lawn is the value of the time taken at the wage rate.

If however, the worker's paid work is limited to five days in the week, then it is not paid work that is being given up in order to mow the lawn at the weekend. Although the concept of opportunity cost seems clear enough, its application in this situation can be quite difficult, because it may be hard to identify what exactly has been given up. Suppose (as suggested in the comment by 1muflon1) that an alternative use of the time was to watch a movie. Fleshing out this scenario, suppose that late on Sunday afternoon, the worker was just about to watch the movie, but suddenly noticed that the grass was getting long, and decided that it couldn't be left for another week, so instead went to mow it. So far, it seems that what has been given up is watching the movie. But suppose that, having mowed the grass, they decide to watch the movie in the evening, displacing some other activity.  Rather than a clearly identifiable alternative that has been given up to mow the lawn, there may be a whole series of activities, each of which has merely been slightly postponed to give priority to another.

A context in which economists make use of the opportunity cost of leisure time is the travel cost method for estimating the value of non-market environmental goods such as parks. In this context the value of leisure time has often been taken to be a fraction of the wage, an influential source explaining the basis for such an assumption being Cesario (1976).  Use of a fraction of the (average) wage can be a pragmatic way to avoid the complexity of identifying what an individual has given up, and for aggregation over individuals in different circumstances.

Reference

Cesario, F J (1976) Value of Time in Recreation Benefit Studies  Land Economics 52(1) Feb 1976 pp 32-41