Can the Word of God and God's wisdom be beneficial to an unbeliever? More specifically I'm wondering in the way that we go to the Bible and to God about concerns or issues that we are facing for instruction and wisdom, can we think biblically about another's concerns or issues that they are struggling with in order to help them, and in overall hopes of leading them to Christ?

I'd like to give a non-believing friend some Biblical advice, but I don't know if it will even benefit him (to be honest, I'm not quite sure what I would even say yet).
This is a two-part answer, because your question addresses two things:

I should also make it clear up front that I am one of those unbeliever types.

1) Yes, wisdom from the bible can be beneficial to an unbeliever.

Because wisdom from all sources is beneficial.  As an unbeliever, no book holds a special place.  This means that all sources can be analysed for wisdom.  The bible in particular contains many parables and thought experiments that can be useful regardless of their source; exercises to challenge one's sense of ethics and morality.

But equally, we could read Harry Potter and analyse some of their circumstances and come out with different, but equally valid, wisdom.  Stories of daring and heroism, and in particular those things that the heroes are willing to do or sacrifice in order to achieve that heroism, are always fun to break down and analyse.  Try to work out the ethics of Snape, for example.

But do be aware that, since unbelievers don't share the same foundational concepts, the stories and ethics can come out somewhat differently.  Many of the more foundational stories of the Old Testament -- The Binding of Isaac, Jonah and the Whale, The Testing of Job all being class examples -- and even the concepts from the New Testament -- the vicarious redemption of mankind by Jesus, Hell, etc. -- can be and are interpreted entirely differently.

2) Don't do that.

Even in the rare case that the story you're going to tell is actually appropriate, telling bible stories to a stressed-out unbeliever is going to come across as preachy and will stress your friendship equally if not more.  Feel free to suggest actions taken by those in the stories as possible routes out of their situation -- after all, this is distilled version of the wisdom that is actually imparted to you -- but just telling the story will just likely cause your message to be lost.