From time immemorial, help has been used as both a noun and a verb. It is listed in the OED as both:
1. Make it easier or possible for (someone) to do something by offering them one’s services or resources:
- The action of helping someone to do something:
The etymology for both usages goes through 5th to 11th century Old English, and the cognates in other languages suggest even more ancient roots.
In any sentence, the syntax will determine the usage of a word, and in the first sentence there is absolutely no ambiguity: help is functioning as a verb.
The ambiguity of the second sentence has nothing to do with help. It is the word her that permits the sentence to be parsed in two different ways:
Her can be used as a third person singular pronoun or as a possessive determiner:
[THIRD PERSON SINGULAR]
- Used as the object of a verb or preposition to refer to a female person or animal previously mentioned or easily identified:
- Belonging to or associated with a female person or animal previously mentioned or easily identified:
Parsing the sentence with her as a possessive determiner, help is the head noun of the noun phrase her help with the homework, while her modifies the noun help as a possessive determiner. In this parsing, have takes on the meaning of possess or receive:
They have her help. = They receive Mona's assistance.
Parsing the sentence with her as a third person singular pronoun, her is the object of the verb phrase have her help with the home work, while help is the head verb of the verb phrase help with the homework. In this parsing, have takes on the meaning of cause or arrange:
They have her help. = They arrange for Mona to assist.
Though some would consider the first parsing cleaner and more concise, both parsings are equally correct grammatically. In the end any change in the meaning of the sentence is negligible, so the ambiguity becomes irrelevant for practical communication purposes.