It depends what you're trying to do. Do you want to rank with the homepage? Or do you have posts and pages that target the keywords?
"Static page option" in WordPress makes sense for some websites and strategies, and of course it depends how well you optimize the static page. You'll have more control over it, so you could craft the perfect landing page with all of your targeted text, the best images, videos, links. This could be hand-picked content by yourself, or you could use plugins to display the most popular content automatically (by views, votes, etc.)
"Your latest posts" option is much faster / easier to get started with and it makes sense to use for some websites. The homepage looks fresh, and recent content encourages people to come back later for updates if they like it. This style is more about ranking the posts for your targeted words/phrases.
But you don't have to choose. You can have half static and half latest posts, or any combination really. For example: a section at the top - "News" - with 5-10 of the latest posts automatically placed there, and below that can be all static.
What is your website about and what style makes sense for that? You've seen other websites before, so compare your idea to similar things out there.
You mentioned onsite optimization, but it's going to be difficult to rank for any competitive keywords with just onsite optimization on a 3 month old website. It's probably best to concentrate on creating content right now. Quality content that serves a purpose other than just having keywords in it.