Background
I have been spending a lot of time in my research with subsets of groups that are very close to being generating sets. To make this precise:
Let $G$ be a group. If a subset $S$ of $G$ projects onto a generating set of $G/[G,G]$, we say that $S$ weakly generates $G$. The following fact (see page 350 in this book for a proof) shows that weak generation in nilpotent groups is a strong condition.
Fact. Let $G$ be a nilpotent group. If $S$ weakly generates $G$, then $S$ generates $G$.
In light of this result, we ask the following question:
Does there exist a finitely presented but non-nilpotent group $G$ such that every weakly generating subset of $G$ generates $G$?
If we drop the condition "finitely presented" then the first Grigorchuk group suffices. I'd be pretty surprised if no finitely presented examples exist.
Surface groups and free groups
In response to Matt's question below: For the free group $F(a,b)$, the set $\{a[[a,b],a],b \}$ weakly generates but doesn't generate (you can show this directly using uniqueness of freely reduced word form in a free group). You can use this to show that any non-abelian closed surface group has subsets that weakly generate but don't generate. For instance, in the genus two case, suppose $G$ has the standard presentation with generators $a,b,c,d$ and relation $[a,b][c,d]$. Consider the set $S = ${$a,b[[b,c],b],c,d$}. If this set generates G, then it generates the image $G/N$, where $N$ is the normal subgroup generated by $a$ and $d$. This image is a free group generated by the images of $b$ and $c$. The set S projects to a set which does not generate.
Best Answer
Here is a little explanation for why nilpotent groups are the only possible example for groups satisfying sufficient finiteness properties. Finitely presented is barely a finiteness property at all, so certainly non-nilpotent examples may exist.
There are no finite non-nilpotent groups such that every weakly generating set is a generating set. If G is such a group, then [G,G] is finite and the condition is simply that the elements of [G,G] can be removed from any generating set of G. This is simply the condition that [G,G] ≤ Φ(G), where Φ(G) is the intersection of the maximal subgroups of G, also known as the Frattini subgroup. A finite group is nilpotent if and only if [G,G] ≤ Φ(G) (and of course [G,G] ≤ Φ(G) for any nilpotent group, finite or not), so this explains why nilpotent groups are important for this property.
Groups G in which [G,G] is noetherian and every subgroup has the property that every weakly generating set is a generating set have a different name: these are the groups in which every subgroup satisfies a very weak form of subnormality, known as being "serial" (sort of like being a member of a composition series whose order type is a fairly arbitrary linearly ordered set). This is described in section 12.4 of Robinson's Course in the Theory of Groups.
Finitely generated soluble groups have well behaved Frattini subgroups, so that if [G,G] ≤ Φ(G), then G is nilpotent. Assuming [G,G] is noetherian, a finitely generated soluble group G in which every weakly generating set is a generating set is nilpotent. In particular, polycyclic groups will not provide any examples.