Something noisy is a good idea. Ideally put it in after he has gone to sleep so it isn't there to distract him from settling but if he only sleeping part of naptime most days then he would play less on tired days before falling asleep and longer on other days meaning he should be waking up at the same time on all days.

What time does the child come in the morning? Is it possible he is sleeping longer on some days then others and therefore his need for nap isnt' consistent.

Between 18m - 2yrs when I think the child is getting near the climb out phase or is sleeping consistently for the time then I move the playpen down to where the older kids are sleeping so they get used to the naproom (mainfloor play area aka was diningroom). Then when they have the process I will try them on a mat like the others with the threat of be a big kid too or back to the playpen for you. Also means I am close by if he gets up and I can reinforce the lay down part without having to run up to the bedrooms or waste naptime standing up there.

Have they made the switch from crib to bed at home and is the child attempting to crawl out of the crib? Do the parents ever let him nap on the couch or their bed or other bed on weekends?