How we cite our quotes:
(Orestes): "Hermes of the Underworld, watching over paternal powers! I ask you, be my saviour and my ally; for my coming to this land is my return from exile." (Fragment 1; 1-3)
The Greek gods all had different, specialized functions. That isn't to say that one god couldn't play many different roles. In fact, this was usually the case. Thus, when Orestes calls upon Hermes and asks for his help, this is appropriate for many reasons. First of all, Hermes is the patron god of travelers and has perhaps been guiding Orestes on his "return from exile." Also, Hermes is the messenger god and the god who brings the souls of the dead down to the Underworld; thus, he can be counted on (if anyone can) to get a message down to his dad Agamemnon to let him know what's up. Finally, Hermes is also the god of trickery and deception. This will come into play later on, when Orestes uses deception to sneak into the palace of Argos and get his revenge.
(Orestes): "O Zeus! Grant me vengeance for my father's death! Be my ally if you will!" (18-19)
A few lines later, Orestes calls on another god for help. This time it's Zeus, king of the gods, who watches over justice. By asking Zeus for vengeance, that's another way of Orestes making sure that his actions are justified.
(Electra): "As I pour these libations to the dead, my own words too call upon my father: have pity on me, and kindle our dear Orestes as a light in the house! […] That Orestes may come here through some fortune is my prayer to you; and you must hear me, father! Grant me also to be much more chaste than my mother, and my hands to have greater piety." (129-131, 138-141)
In the ancient Greek world, the spirits of prominent individuals were sometimes worshipped after they died. These dead famous dudes were known as "heroes" and were usually associated with a particular location; the practice of worshipping them is known to modern scholars as "hero-cult" or, simply, "hero-worship." Here, Electra makes prayers to the spirit of her father Agamemnon as if he were a hero watching over Argos. Just as in a prayer to any other god, she asks him for help in this world.