A counterexample to this question can be constructed as follows.
Let $X=[0,1]$ be the closed interval with the standard Euclidean topology.
Let $Y=\omega$ and $\mathcal T_Y$ be the topology on $Y$ consisting of the sets $W\subseteq Y$ satisfying two conditions:
$\bullet$ if $0\in W$, then $W=\omega$;
$\bullet$ if $1\in W$, then $\omega\setminus W$ is finite.
The definition of the topology $\mathcal T_Y$ implies that $\omega\setminus\{0,1\}$ is an open discrete subspace of $(Y,\mathcal T_Y)$.
It is easy to see that the space $(Y,\mathcal T_Y)$ is not locally connected at $1$.
Choose any sequence $(U_n)_{n=1}^\infty$ of pairwise disjoint nonempty open sets in $X=[0,1]$ and let $g:X\to Y$ be the function defined by
$$g(x)=\begin{cases}n&\mbox{if $x\in U_n$ for some $n\ge 1$};\\
0&\mbox{otherwise}.
\end{cases}
$$
The definition of the topology $\mathcal T_Y$ ensures that the function $g:X\to Y$ is continuous.
Let $C=\{(x,g(x)):x\in X\}\subseteq X\times Y$ be the graph of the function $g$. It is clear that $C$ is homeomorphic to $[0,1]$ and hence is compact, connected and locally connected. But the projection of $C$ onto $Y$ is not locally connected.
Best Answer
One could phrase local compactness using neighbourhood bases as well (in the Hausdorff case, at least) if desired: once one has one precompact open neighbourhood, one automatically has a whole neighbourhood base of precompact sets, since any subset of a precompact set is still precompact. (And in practice, this is often how local compactness is actually used.)
The most important thing that a property named "locally P" should obey is that it be local rather than global. In the case of topological spaces, this means that if a topological space X is locally P at some point $x_0$, and we have another topological space Y which agrees with X at a neighbourhood of $x_0$ (e.g. Y could be the restriction of X to a neighbourhood of $x_0$, with the relative topology), then Y should be locally P at $x_0$ as well. Both local compactness and local connectedness, as defined traditionally, have this locality property. On the other hand, the property "$x_0$ has at least one connected neighbourhood" is not local (consider for instance $({\bf Q} \times {\bf R}) \cup \{\infty\}$ in the Riemann sphere ${\bf R}^2 \cup \{\infty\}$, which is a globally connected space which is locally identical near the origin to ${\bf Q} \times {\bf R}$ in ${\bf R}^2$, which has no connected open sets), and thus this property does not deserve the name of "local connectedness at $x_0$".
It may help to think of the modifier "locally" not as a rigid recipe for converting global properties to local ones, but rather as an indicator that the property being modified is a local analogue of the global, unmodified, property. In most cases there is only one obvious such analogue to select, although in some cases (such as the notion of "locally compact" in the non-Hausdorff setting) there is some freedom of choice, which ultimately means that one has make a somewhat arbitrary convention regarding terminology at some point.