On-page optimization, or link-building?

If this is a content site.... neither.... Spend the time to make social sharing very easy.    Any left-over time trying to get links from hub sites in that niche.

If this is a retail site... would go for links from the easy industry and community resource sites.

For both sites would ask for 8 more hours to brainstorm highly linkable content that the site owners will produce.

In the question between a subfolder and a subdomain...  it is clear and it is murky...

Search engines say things and often do the things that they say.  They also do things without sayin' that they did them and you are left to figure out what happened or how things work on your own.

If you have been running websites for a long time (24 years here), you might know that Google and other search engines have treated subdomains and folders differently,  identically and murkily.   It depends upon what year and month you are talking about.

**The constant is that subfolders have always inherited ranking ability from their parent domain, and will likely do that into the future.  **

So, if you want assurance into the future, using a subfolder is your best bet.

Looking at DA... it is irrelevant in the subdomain and subfolder discussion.  DA is a moz number that kinda correlates with search engine rankings, but just part of the time.   Ranking ability and DA are different things.

I watch analytics on a website (for a friend's business) that is reasonably stagnant, which just experienced a massive spike in search traffic for no explainable reason.

Try digging a little deeper into the analytics.

If you are looking at his analytics you should be able to see where those visitors were coming from on that day... also pages that they entered and lots more.   The answers are just a few clicks away.