What does the Noble Eightfold Path say about right speech when it comes to persuasion and influence?
While persuading and influencing are not part of the standard definition of Right Speech, it certainly is considered inappropriate for monks and not part of Right Speech for monks, in the context of monetary gain or politics or other worldly endeavours not suited to the holy life, as seen below.
From DN 2:
"Whereas some brahmans and contemplatives, living off food given in faith, are addicted to running messages and errands for people such as these — kings, ministers of state, noble warriors, brahmans, householders, or youths [who say], 'Go here, go there, take this there, fetch that here' — he abstains from running messages and errands for people such as these. This, too, is part of his virtue.
"Whereas some brahmans and contemplatives, living off food given in faith, engage in scheming, persuading, hinting, belittling, and pursuing gain with gain, he abstains from forms of scheming and persuading [improper ways of trying to gain material support from donors] such as these. This, too, is part of his virtue.
Right Speech is abstinence from Lying, Harsh Speech, Empty Speech and Divisive Speech. If one is ordained then abstaining from 32 types of talk are also required.
If one persuades someone to the Dhamma one does an immense meritorious act. If one uses one's influence in order to make people listen and practice the Dhamma (Eg: A boss making his employees follow a Buddhist meditation or listen to Dhamma Talks) one again does a immense meritorious act. So it will be in accordance to right speech to use persuasion and influence in order to bring one towards the Buddha's Teachings.
But if one uses persuasion and influence in order to increase his own standing (even among Bhikkhus) or to anything that is not conducive to one's elimination of suffering, one is not following "Right Speech".