Is there a term for grammatical mistakes as a result of trying too hard?

Hypercorrection

It can even influence a language: Fetus is the original Latin term, but it got "fixed" to foetus or fœtus both in how people wrote Latin, and also in languages that had a term derived from it so now foetus is sometimes found in English, Dutch and German and also fötus in German; all based on people "fixing" what wasn't broken.

In between personal hypercorrection and something introducing a new spelling that became common across four different languages, are myths like "the passive voice should be avoided", "don't start a sentence with a conjunction", "don't start a sentence with however", "don't split infinitives", "don't split any verb form", "don't end a sentence with a preposition", "don't use the genitive of an noun that represents a non-living thing", "don't use like as a conjunctive". Here not only has someone got a bizarre notion into their heads about what is correct, but the "rule" has ended up being actively taught.

Often it's combined with an incorrect understanding of its own terms, as in this notorious example: source

Not only is this teacher enforcing an imaginary rule, but of the 10 cases marked as "P.V.", only three are in the passive voice. Of those 3, only one seems like it could be rewritten into something that didn't weaken it (and that one still wouldn't actually improve). One actual use of the passive voice isn't marked as such.

In theory, such a "rule" could become so popular, that it became a real rule just by dominating the thinking of those who spoke the language. Many though are actually impossible to follow consistently, or just too at odds with common literate use, to ever reach that point.

Hypercorrection.

Wikipedia has:

In linguistics or usage, hypercorrection is a non-standard usage that results from the over-application of a perceived rule of grammar or a usage prescription. A speaker or writer who produces a hypercorrection generally believes that the form is correct through misunderstanding of these rules, often combined with a desire to seem formal or educated.

Linguistic hypercorrection occurs when a real or imagined grammatical rule is applied in an inappropriate context, so that an attempt to be "correct" leads to an incorrect result.

Hypercorrection is sometimes found among speakers of less prestigious language varieties who produce forms associated with high-prestige varieties, even in situations where speakers of those varieties would not. Some commentators call such production hyperurbanism.

I've seen this kind of mistake referred to on this very site as a hypercorrection:

The use of a nonstandard form due to a belief that it is more formal or more correct than the corresponding standard form.

You've already got your answer, but a related term is "superstandard" or "supercorrect" speech which is perceived by most people as going beyond the requirements of normal language use but not necessarily going so far as to break grammatical rules.

From http://www.linguistics.ucsb.edu/faculty/bucholtz/articles/MB_JLA2001.pdf:

A linguistic superstandard is a variety that surpasses the prescriptive norm established by the standard.

and,

I use the terms superstandard and supercorrect to distinguish between true hypercorrect language use (that is, use that violates a rule of descriptive grammar) and strict adherence to prescriptive grammatical rules accompanied by the use of other formal language features. As discussed below, however, hypercorrection is often part of superstandard English.