A participant dies. The benefit is then of course payable to their named beneficiary (assuming they completed a beneficiary designation while they were living naming some other living person). Can this named beneficiary then in turn name their own beneficiary for the benefit that they are entitled to? If so, can it be any beneficiary, or can it only be a spousal bene? Does this have to be provided for in the plan document?

Depends on the type and timing of the permitted distribution options payable to the beneficiary as mandated by the plan doc. Remember, minimum required distributions must be paid (pursuant to the terms of the plan) over a specified time period following the participant's death. So, if this is a DC plan mandating a full cash distribution within 5 years of death, then the beneficiary could name another beneficiary to take if the participant's beneficiary dies within that 5-year period. But, if the beneficiary must take in the form of an annuity, whether a beneficary can be named for those annuity payments depends on the document and the nature of the annuity.

As stated, it depends.
Here is another example. Annuity payment form; retiree elects LA with 10-yr certain, then retiree dies within 10 years. Somebody gets the remaining payments, according to the beneficiary election (or plan default if needed). That somebody can (and should) name a beneficiary in case she/he dies before all remaining payments are made.