A member of my family is an atheist hostile to faith and likes to challenge me to explain how a loving God permits suffering. Is there an answer I can give?

It is perhaps too absolute to say the Church has "an answer" to suffering, and why God permits it. The full answer to this is ultimately mysterious, and has many aspects, which are hidden from our view. 
 
To be sure, we do have elements of an answer, which God reveals to us, or which human reason can supply. One element of an answer is the existence of human freedom. One way God could prevent a lot of suffering would be to cancel human freedom whenever it was abused. But God, it would seem, does not usually see fit to do so, and his respect for our freedom is very consistent. 
 
Another aspect of an answer is that suffering often brings growth and opens new possibilities. Perhaps God sees these fruits and thereby allows some degree of suffering. 
 
But again, these insights are part of an answer. They do not constitute a full or complete answer to the great mystery of suffering and why God allows it. These insights tend to bring up even more questions. Fundamentally we must accept that we do not have an absolute answer to the problem of suffering. 
 
While it is fine that atheists raise this issue, (for it is a valid question we all have), the demand by them for an absolute answer is not reasonable. We must also ask them to consider that not everything has a simple answer. 
 
Thus, if they will demand that I must absolutely answer the question, “Why is there suffering?” then I would also like to ask them to give an absolute answer to the question, “Why is there love? Why is there generosity or a passion for justice or self‐sacrificing heroism?” 
 
If you and I must account for the negative side of suffering, then perhaps too our critics must account for the existence of love. If they are honest they will perhaps admit that while they can give a partial answer, they cannot fully account for these things. 
 
The bottom line is, not everything can be absolutely answered. To unreasonably demand answers from others, when they cannot supply them either, is not itself a rational or adequate reputation of the existence of God.