You've gotten a few things confused here.
- Here, ふる is a verb meaning "to dump" (or "to reject"). I don't know that "trip up" is a meaning of ふる, though I could well be unaware of it. So, ～にふられる means "to be dumped by ~".
- The 恋人 is the 友達's 恋人, not the speaker's.
- 落ち込む should be seen as a single verb with its own meaning here, rather than as a compound of 落ちる + 込む (indeed, many verbs that are [stem form] + 込む should be viewed as one verb rather than two). In this case, 落ち込む means "to feel down". (Also, こむ as in "to be crowded" is more commonly written 混む rather than 込む.)
- Here's the answer to your main question: when 一体 is used at the beginning of a question, it makes the question more emphatic. (See also: Why does 一体 mean "what the heck?"?)
- [て form] + あげる means "to do ~ for / on behalf of / as a favor to someone". Similarly, you can use [て form] + やる for "to do ~ for an inferior" and [て form] + [差し上げる]【さしあげる】 for "to do ~ for a superior". So, 言ってあげる means something like "to say something as a favor to my friend" (but this sounds terrible in translation).
- I think it is better to translate 全然分からない here as "I don't understand" rather than "I'll never understand" (present rather than future).
Putting all this together, your translation should look more like this:
My friend was dumped by his lover, and he's really dejected. But, I really have no idea what to say to him [e.g. to make him feel better].
I'm not sure what the best way to translate the nuances of ～てあげる is, but the point is that you're not just saying something to him; you're saying something to him with the intention to help him out.