I'm a Lebanese American and my dad always says " We aren't Arabs, we're Phoenicians."

I am going to guess you're from a Lebanese Christian family. I wrote about the ethnicity of Syriac Christians here. You can skip down to the "Are they really Assyrian?". While it only address the ethnicity of Lebanese Christian in one sentence, many parts still apply. The sentence is:

This is why Maronites of Lebanon often call themselves Phoenician: they may speak Arabic now, but they are not descended from the Arab invaders.

The one caveat I would say is that before the Arab invasions, I'm doubt the local Christians thought of themselves as "Phoenicians"; rather, I get the sense they thought of themselves as "Romans" (that is, Eastern Romans, that is Byzantines) [edit: see down the thread for clarification of this].

But the issue gets at the core question of what is an ethnicity? Benedict Anderson defined a nation like this:

In an anthropological spirit, then, I propose the following definition of the nation: it is an imagined political community - - and imagined as both inherently limited and sovereign.

Full context of his quote defining every word but "political" here. That definition, nationality/ethnicity as an "imagined community", is the most common definition used by historians and social scientists.

The question is, how do you imagine ethnicity in this particular case? If it's defined by language and culture, as many of the Arab nationalists had it (several prominent Arab nationalists of the 1950's and 1960's were Christians), it clearly included Christians and to some extent Jews. If it's defined by common descent, the Christian groups probably can make a good case for not being Arabs, as there was historically little intermarriage between Christians and Muslims (or, if there was, the children were generally raised as Muslims). Where this meme of Lebanese Christians as Phoenicians comes from, I'm not sure, but I would guess it's meant as an answer to Pan-Arabism that swept the region in the 1950's and 1960's. For a book on Pan-Arabism, I like Adeeb Dawisha's Arab Nationalism in the 20th Century: from Triumph to Despair. First chapter is available as a free PDF online, but I don't remember precisely what he says about Arab Nationalism and Lebanese Christians.

He says a lot about Syrian and Iraqi Christians, who generally engaged actively in Pan-Arab movements. To understand why Lebanese Christians were more reluctant, it's important to recall that Lebanon was basically carved out of historic Greatness Syria (Sham) as a Christian stronghold (Mount Lebanon of course has long had a special status in the Ottoman Empire as Mustariff since 1861, and other status before that, but that's getting very far into the past and generally only include a portion of modern Lebanon, but obviously the most Christian parts). Lebanon was supposed to be a place where Christians were in the majority, and so Lebanese Christians were reluctant to give that up in favor of Pan-Arabism.

Ah, actually, of course there's a Wikipedia page on the subject: "Phoenicianism". That says the idea came slightly earlier (in the 20's rather than the 50's), but in the same context I suspected: creating a national identity for Lebanese Christians separate from their identities as Ottomans, Arabs, Syrians, or even Greeks. It's not a very good Wikipedia article, however. This sort of nationalism emphasizes all the common tropes: we are the original inhabitants, we are pure, we have a past glory that can be returned to, our ethnic rivals are recent comers have no claim to the area, etc. Compare it to Zionism, Palestian nationalism, Turkish nationalism (which once claimed descent from Hittites and Sumerians to be able to say they were in Anatolian before the then rival Greeks, but still with the traditional nationalist view of Kurds as "Persianized Turks who forgot their roots"), Serbian nationalism (especially around areas like Albanian-majority Kosovo), whatever.

That doesn't mean it's completely without merit in this case. If we go by descent, there's a good argument that there is a notable portion of descent that they do not share with neighboring Muslim Arabs (though obviously, most neighboring groups are genetically similar because people don't move around that much). What to call that is hard to say. Before the Arab Invasions, the residents of that area were Byzantine (Greek-speaking Roman) Christians, but they had arguably only been Christian for a few centuries between roughly Constantine and the Arab invansions. Before that, they were... well, part of a series of different Empires, including the Persians, Greek, Seleucid, and Roman Empires. Throughout this period, it seems many of the local pre-imperial gods continued to be worshipped (alongside new gods). And before that, Phoenicians.

Does that make Lebanese Christians today ethnically Phoenicians? If you go by descent, and not language or culture or something along those lines, there is an argument to be made here. Whether you find it convincing is up to you. Are Arabic-speaking Christians (particularly Lebanese Christians) their own ethnicity or are they just a particular sort of Arab? There's no scientific answer to this question, anymore than there's a scientific answer to whether my ancestors who were German-speaking Jews were ethnically Jewish who happened to speak German or ethnically German who happened to follow the Jewish religion. My grandparents saw themselves as Germans who happened to be Jewish, but of course Hitler saw them as Jews who happened to speak German. It really comes down to how you (and others around you) "imagine" the limits of the relevant ethnic communities. Rogers Brubaker, in his Citizenship and Nationhood in France and Germany, emphasize that some places, like France, insisted on a culturally-minded "civic nationalism", while others, like Germany, insisted on a descent-minded "ethnic nationalism". You continue to see both kinds, and lots of room between, in the world today.