The son, who is in his late thirties, has recently become discouraged and has stopped attending church services. Since the New Testament does not address this concern specifically, one must attempt to draw a reasonable conclusion based upon the evidence available. My personal view is that the defection of an adult son, in the circumstances you describe, does not necessarily disqualify the brother from serving as an elder. The elder may have raised his children in the ways of the Lord, but then later, when the child was on his own, events transpired wherein the offspring made the individual decision to abandon faithful service.
Every young person is guilty of sin.
It is not reasonable to expect that the elder's home will be absolutely free from the common problems that all Christians experience in rearing children. Having said all of the above, we must note this. This need not be the case necessarily, but it could be. The judgment of the local church should be consulted in such a case. A good elder will always put the welfare of the congregation foremost in any decision.