Sure, mechanical/non-magical equivalents would be a fluffy way to satisfy as aspect of his role-playing. But how about you try something interesting?

Word of your superstition and abstinence against magical gear has spread and intrigued a number of people. You've awakened old fears of harmful magical auras, and new prides in personal strength. Scholars and philosophers start sending you mail asking you questions or for instructions in how to live as a 'magi-Luddite' as the trend is being called. You become a minor celebrity with people asking to follow in your footsteps.

In short, give him the leadership feat. With the string that if he started using magical gear (in public) his following would abandon him.

First, answering the question as posed: No, there are no rules (actually guidelines) about it, at least I couldn't find any in either the DMG or XGtE, where most of the general guidelines are found.

This is not a problem, though, because when you are DMing, you have two major options: Either you are running a premade campaign or you are running a homebrew campaign. In both cases, you problem can be solved.

Premade Campaign

Most published adventures state the magic item itself, rarely telling you to roll for it. Usually you won't have a problem there. If you do, handle it the same as the homebrew campaign below.

Homebrew Campaign

This is your world. When you wrote this:

Just making the decision myself about what they find, however, feels quite arbitrary.

IMO this is just wrong. You are the world designer. Deciding what monster will be there, how they behave and what treasures the PCs will be rewarded is part of your job, or, if you prefer, is part of your powers. You might think it's arbitrary if you don't put too much thought into it, though, which leads us to...

Give life to the magic items

The DMG has some suggestions on how to create magic items. Check p. 141 for my point.

You can add distinctiveness to a magic item by thinking about its backstory. Who made the item? Is anything unusual about its construction? Why was it made, and how was it originally used? What minor magical quirks set it apart from other items of its kind? Answering these questions can help turn a generic magic item, such as a +llongsword, into a more flavorful discovery.

But don't stop there: Why is the item here? Why is that +1 Longsword, that now we know it's actually the sword of Percival, the Holy Knight of the Kingdom, which was blessed by the High Priest 300 years ago, lost in a freaking Orc Cave? Did they raid Camelot recently and got it as a spoil? The feeling of "arbitrary decision of putting it here" will be nothing to you or your players when you can put a background there, when the item has a plot and simply makes sense for it to be there. That lore can be created before the characters, so you are actually unbiased and not making a world tailored for them.

Obviously be careful with this line so you don't give them too many useless items (for them). But this shouldn't happen in a fairly balanced party - someone will be able to use that staff, dagger and sword.