Is there a name for this type of logical fallacy? Consider a statement of the form: $A$ implies $B$, where $A$ and $B$ are true, but $B$ is not implied by $A$. 
Example: As $3$ is odd, $3$ is prime. 
In this case, it is true that $3$ is odd, and that $3$ is prime, but the implication is false. If $9$ had been used instead of $3$, the first statement would be true, but the second wouldn't, in which case it is clear that the implication is false.

Is there a name for this sort of logical fallacy?

 A: The implication is true. A related sentence, "$\forall n$ if $n$ is odd then $n$ is prime" happens to be false.
A: I think that would just be a non sequitur ("it does not follow"), which doubles as a catch-all term for all invalid arguments.
From Wiki:

Non sequitur (Latin for "it does not follow"), in formal logic, is an argument in which its conclusion does not follow from its premises. In a non sequitur, the conclusion could be either true or false, but the argument is fallacious because there is a disconnection between the premise and the conclusion. All invalid arguments are special cases of non sequitur.

In your case your premise and conclusion happen to be true, but B does not follow since the implication is broken. 
A: Affirming the consequent
This fallacy takes the following form:
Premise 1:If A, then B.
Premise 2:B.
Conclusion:Therefore, A.
Argument: "Those who practiced authorized plural marriage had multiple sexual partners. John C. Bennett had multiple sexual partners. Therefore, Bennett practiced authorized plural marriage."  
Rebuttal: A implies B, but B does not imply A--i.e., authorized plural marriages had multiple partners, but all those with multiple partners were not practicing authorized plural marriage.
A: If $P(x)$ and $Q(x)$ are statements about $x$, what happens is that $\exists x_0$ such that $P(x_0)$ and $Q(x_0)$ are true, then you are concluding that $\forall x$, $P(x)$ and $Q(x)$ are true, which is not necesssarily a valid conclusion.
A: Apparently, it is called affirming the consequent.  It is also referred to as a converse error or the fallacy of the converse.
A: This is not strictly a fallacy; this is a result from the truth-table definition of implication, in which a statement with a true conclusion is necessarily given the truth-value true, i.e., for any formula $A\rightarrow B$ , if B is given the truth-value T, then, by the truth-table definition of $ \rightarrow $, it follows that $A\rightarrow B$  is true. The only invalid statement of this sort is a statement of the sort
$A\rightarrow B$ in which A is true and B is false; this is the essence of what logic is about; we want to avoid starting with a true statement, arguing correctly, and ending up with a false statement; logic is largely designed to avoid this, as correct reasoning is that which preserves true premises, or sends true premises to true premises.
A: Others have said this is not a fallacy, and that is true in the logics we are used to using in mathematics, but perhaps not in the logics used in natural language or, say, the law.  For example, Relevance Logic is designed to disallow this sort of thing.
