My wife (Polish passport) is planning to travel to the USA (and some European countries) with our son who has dual Polish-British nationality. Would it cause any issues if the young one is travelling on a British passport? It seems especially convenient this way, because Polish citizens still need a visa to get to the USA, which is both expensive and humiliating to obtain. Are they going to experience any issues with American officials?
Technically there's no issue at all with doing this.
However where she could have a problem is in convincing the immigration officials that she is the child's true parent, and has authority to take him across country lines. I'd suggest taking a copy of your son's birth certificate showing that she is the child's mother, as well as a letter from you giving permission for her to travel with him. If her name on the passport is different to that on the birth certificate, such as if it changed at marriage, you'll also want proof of that - such as a marriage certificate.
Here's some additional information on children traveling with only one parent.
Don't forget that you will be required to obtain an ESTA for your son to enter the US under the visa waver program.