Bsrgvty

Does mean(s) modify a singular state of having fewer errors, or the plural errors themselves?

I think you are correct that the grammar can go either way. Personally I think the more logical answer in this case is the first version you list ("fewer errors means better products"). I interpret it as having two understood gerunds, and therefore it can be expanded to:

"Making/having fewer errors means producing better products."

Grammatically, I would go with the second statement. However, both statements are correct. Check this out:

“Fewer errors means better products”

Here you are treating "fewer errors" as a single entity that will then result in better products. Due to that singularity treatment the statement is correct.

"Fewer errors mean better products"

In this statement, you are treating "fewer errors" as separate errors hence inferring plurality. Due to that the statement is also correct.

One little trick that i like to play is try using "these"/"this" in either statement to see which brings out the grammatically correct statement. For instance:

In the case where we have treated "fewer errors" as a singular concept, it will be:

"This fewer errors means better products”

In the case where we have treated "fewer errors" as a plural concept, it will be:

"These fewer errors mean better products"

Going from the above, I believe the latter is a very natural way of saying it, and I would go with it in my statement.

To understand more about subject-verb agreement, here are some helpful resources:

Subject-Verb Agreement

Making Subjects and Verbs Agree

Agreeing with IanM, both are grammatically correct to my ears as a native US English speaker.

The two phrases are very similar in meaning, but they have slight nuances that distinguish them. As IanM points out, the singular "fewer errors means..." treats them all as one entity while the plural "fewer errors mean..." treats them as many entities.

What I'd like to add to IanM's answer is how those two cases would be treated differently. If I hear the plural version, "fewer errors mean..." then I will be looking at each error independently, and I will be looking for each error to provide something tangible which makes the whole product better. If I hear the singular version, "fewer errors means..." then I will be looking at the errors as a whole. It would be okay to fix an error and find it didn't make the product better, because I am looking at all the errors as a whole.

I am a programmer by trade. If, as a programmer, I heard the plural "fewer errors mean..." it would lead me to go look at the list of known bugs in my program and see if I can squash a few of them. However, if I hear the singular "fewer errors means..." I would look more at procedural tools which are good at catching bugs in general, like unit tests or peer reviews.

In 3rd grade US English class in 1958 the rule taught was, "If the noun is plural the verb is not". So "Fewer errors mean better products" is the correct form based on that rule taught at that time.

The rule served me well through college as the English instructor could be heard ranting every other day about the noun and verb matching in submitted essays.

The second form is more elegant in the OP's case, as the term "fewer" implies countability, in which case "mean" is the correct conjugation. If it was a non-countable quantity, "means" looks more correct to me, eg "High phosphate in rivers means explosive growth of phytoplankton and algae", or "increasing poverty means increasing discontent", as opposed to "a high population of non-vaccinated individuals mean that risk of an epidemic is increased" (population being countable).

However, in all of these cases I feel the word "mean(s)" is not the right one, and implies a political stance, an opinion, or a dogma, and perhaps condescencion. My experience as a Brit that prefers the Oxford style of written English that "lead to" would be better in the OP's case, or indeed "causes", "increases the likelihood that".

There is a subtle difference in meaning, between the two. Both are grammatically correct, the choice depends on what you mean to say.

‘Fewer errors means better products’ is generic - like ‘business advice’ that you might give someone.

Eg
"At Smiths’ sprockets, we use quality control practises, because we believe that ‘fewer errors means better products’. The ‘means’ refers to ‘fewer errors’ here. The process of how you do something.

The second one ‘Fewer errors mean better products’ focuses on the products. ‘Mean’ refers to ‘products’. The meaning is thus about the result of what you get out of it.

Eg ‘by using high quality beans, and keeping out the bad ones, our baked bean product tastes better - because fewer errors mean better products’.