I've been combing through the things in my Player's Handbook that I normally gloss over and realized that there's not much distinction between studded leather armor and leather armor. The 5e manual states about studded leather armor:

Made from tough but flexible leather, studded leather is reinforced with close-set rivets or spikes. [...] Weight: 13 lbs

And about leather armor:

The breastplate and shoulder protectors of this armor are made of leather that has been stiffened by being boiled in oil. The rest of the armor is made of softer and more flexible materials. [...] Weight: 10 lbs

So to my understanding, either leather armor's boiling in oil is something that studded leather armor doesn't have, or the only difference between the two is about 3 lbs of rivets/spikes.

Comparing the price per weight to things like ball bearings, or 5 lbs of iron spikes, the rivets/spikes used in the armor should come out to about 1 gp.

If you had a party member who was proficient in leatherworking and was willing to donate their time, could you transform a 10 gp Leather Armor into a 45 gp Studded Leather Armor for a mere 1 gp?

Not by Rules as Written.

The Downtime Activities section of the DMG has nothing on creating mundane items, just magical ones.

The optional Downtime Revisited rules from Xanathar's Guide to Everything, however, have this to say (XGtE 128):

Resources and Resolution. In addition to the appropriate tools for the item to be crafted, a character needs raw materials worth half the item's selling cost.

Therefore even if you counted Leather Armor as raw materials, you would only have 11 gold out of the required 22.5 gold.

The XGtE Downtime Revisited rules are optional, of course, so a DM could just rule that the plan works anyway, especially in a place like this, where it can be reasoned.

Additionally, below this is a table that states the tools required for various armors. It lists leather armor as something leatherworker's tools are capable of, and all other armors are made using smith's tools. It is unclear as to whether "leather armor" here applies to both types of leather armor or just the stat block of Leather armor, but I would rule that the creation of studs and rivets requires the use of smith's tools because of its metal nature.

RAW, Leather Armor cannot be upgraded to Studded Leather Armor

I think it's reasonable, given that all the items involved are relatively low-tier equipment for the game, for a DM to fiat a method of cheaply upgrading one kind of armor to another. But the Rules-As-Written don't provide any such mechanic. It only provides rules for purchasing said equipment, or crafting the equipment wholesale for resources worth half the normal selling price with the benefit of the proper Artisan's Tools.

Using the price of Iron Ball Bearings to predict the price of Iron Rivets/Spikes is not a good idea

In real life, two objects that have the same mass and material type can still have very different properties [and consequently, cost] based on how they're manufactured. 5th Edition D&D doesn't spend a lot of effort trying to hash out these complications, and simply sets prices for standard objects and handwaves over the possible contradictions.

So neither the PHB nor the DMG attempt to price out the individual rivets used to produce Studded Leather Armor. But since Studded Leather Armor costs 35gp more than Leather Armor, and since a character with Leatherworking tools proficiency is able to produce a set of Studded Leather Armor at the cost of 22.5gp (using either the PHB rules for crafting mundane items or the Xanathar's Guide to Everything rules for crafting mundane items) we have to presume that between the leather of the armor itself and the rivets, 17.5gp of resources must be accounted for (the difference between crafting Leather Armor at 5gp of its normal 10gp price, and Studded Leather Armor at 22.5gp of its normal 45gp price). Again; the rules don't stipulate on exactly what that value constitutes.

Maybe Studded Leather Armor also has a higher quantity of Leather than regular Leather Armor, or maybe the rivets really are that expensive. Either way, it makes it unlikely that in the diegesis of 5th Edition D&D, the price of 5lbs of Iron Rivets is only a single gold piece; or at the very least that it's accurate to pin its price as such based only on a comparison to Ball Bearings.

The time a crafter spends on their craft is worth money, too. The 1gp for the studs, the 10gp for the armor, and the time working on it fills out the remaining cost requirement. (Time is arguably not a 'material' but it is 'material' as in 'relevant' to the creation of the item.) So if RAW fail to support good sense, then RAW is in the wrong, and should be overruled.