The topics of the Young measure and varifold theory popped up in Friz and Hairer's text on rough paths as deterministic analogues of the machinery being developed there.
I'm looking for reasonably thorough references on these two topics that will, ideally, make me feel more acquainted with the project of Friz and Hairer, and their overall project with renormalization.
On
The other answer addresses important questions that the asker might want to consider. I will stick to answering the title question which is about understanding “deterministic analogues of rough path theory”. I am afraid that there is a significant confusion here. Rough path theory is an entirely deterministic theory. It is a theory that provides existence and uniqueness results for controlled differential equations, which are of the following kind:
$$ dY =f(Y)dX$$
where the variables are Banach-valued functions and are entirely deterministic objects (just think of a function taking values in $\mathbb R^d$ for a simple example of a Banach-valued function). Look at Levy, Lyons, Caruana (2007) and you will better appreciate that rough paths belongs to the subfield of “Classical Analysis and ODEs”. It is motivated by applications in probability, but there is no probability in the theory itself.
A careful reader would note that Friz and Hairer have written a section in their booj called “analogies with other areas of mathematics“, as opposed to “analogies with nonprobabilistic or deterministic areas of mathematics”.
The book of Fritz and Hairer cites reasonable references on both Young measures and Varifolds. You should try those if you do want to read more about those topics.
I'd like to address some apparent confusion about the relation between the subjects in the question though since if you are interested in either Rough Path theory or renormalisation of singular SPDEs reading these references might not be the best use of time.
The theory of Young measures is similar "in flavour" to rough path theory in that it replaces an object with an enhanced version of itself in order to be more amenable to direct analysis. However, it is in no sense a pre-requisite for Rough Path theory and is not a good use of your time if you want to learn more about Rough Paths (it certainly won't be relevant later to reading the book of Fritz and Hairer).
A similar phenomenon is true for the relation between varifolds and rough paths (here there is a more detailed description in the book so I won't write more).
Both of these examples are relevant because they illustrate that some of the features of the approach of rough path theory (replacing a path with an object valued in a certain tensor algebra and from there constructing the rough integral) are a reasonable extension of the classical theory of ODE by analogy to similar approaches in other areas of mathematics. However deeply understanding them will not significantly improve your ability to follow that book.
Finally, I'd note that referring to the book of Fritz and Hairer as part of a project on renormalisation is potentially misleading. Renormalisation in this context is primarily a phenomenon appearing in relation to singular stochastic PDEs. Whilst some particular examples can be treated via application of Rough Path theory, most cannot be. If you are interested in renormalisation, you are more likely to be interested in regularity structures (as are briefly introduced in the last sections of that book or in several short introductions Martin has written that are available on ArXiv 1) or in the paracontrolled calculus of Gubinelli, Imkeller & Perkowski (see 2 for some lecture notes by Gubinelli & Perkowski that use some of those techniques).