What is the grammatical function of the infinitive 'to aid them' in the following sentence? 'He could do nothing to aid them.'

The interpretation where the to-infinitival is a dependent of the noun nothing rather than an adjunct of purpose is likely here.

If it were an adjunct of purpose it would tell us why nothing would be done.

He could do nothing.

Why would he do nothing?

To aid them.

However, the salient reading in this case is not that doing nothing would aid them, but that it is not possible for him to aid them by doing any of the things he can do.

A relative clause with the same sense could be substituted:

He could do nothing that would aid them.

Compare:

The students had nothing to aid them with their work.

Again, there are two possible interpretations:

1. If the to-infinitival above were an adjunct of purpose, the students would be doing the aiding. This is however unlikely as the simple fact of the students having nothing is not likely to help themselves or some other group get work done.

2. The salient interpretation takes the to-infinitival as a dependent of nothing so that there is nothing such that the students had this and that this could aid them (the students themselves or some other group) with their work. This is again the relative clause interpretation.

Further examples of infinitival relatives from CaGEL p1067:

i She’s the ideal person [(for you) to confide in ___].

ii I’ve found something interesting [(for us)to read ___].

iii A systems analyst wouldn’t be such a bad thing [(for her) to be ___].

iv That is not a very good way [(for him) to begin ___].

v You're not the first person [___to notice the mistake].

The relativised elements here are respectively complement of a stranded preposition, direct object, predicative complement, manner adjunct, and subject. Except in the latter case a subject can be optionally included, preceded by the subordinator for.