There are spaces, for example the cone of the Hawaiian earring, which are contractible but that have a basepoint such that no contraction fixes that basepoint. Are there any good sufficient conditions which guarantee that a contractible space $X$ with basepoint $x_0$ must have a contraction fixing that basepoint?
One idea I had is, if $(X,x)$ has the homotopy extension property for any $x\in X$, we could take a contraction $H(x,t)$ from $X$ to $x_0$ which doesn't necessarily fix $x_0$ and then compose with a homotopy $G(y,t)$ from the identity taking $H(x_0,t)$ to $x_0$ for each $t$, giving a contraction that does fix $x_0$. The HEP would give such a homotopy for each particular $t$ but the problem is making a choice that simultaneously works for all $t$.
It is not true that the cone $CH = H \times I/ H \times \{1\}$ of the Hawaiian earring $H$ is contractible but does not have any contractions that fix a particular basepoint. Yes, there are basepoints $x_0 \in CH$ which do not admit a basepoint preserving contraction, but for example the tip of $CH$ admits such a contraction.
Now let $X$ be contractible and $x_0 \in X$. You ask when the pointed space $(X,x_0)$ is pointed contractible. Here is a well-known theorem.
Theorem 1. if $(X,x_0)$ is well-pointed (which means that the inclusion map $\{x_0\} \hookrightarrow X$ is a cofibration), then $(X,x_0)$ is pointed contractible.
The proof is not completely elementary. See for example Tyrone's answer to Strong deformation retraction. (he proves a more general result: If a subspace inclusion $i : A \hookrightarrow X$ is both a cofibration and a homotopy equivalence, then $A$ is a strong deformation retract of $X$). Also see Proposition 4.4.14 in
Here is partial converse.
Theorem 2. Let $(X,x_0)$ be pointed contractible. If $\{x_0\}$ is a zero-set in $X$ (i.e. $\{x_0\} = \phi^{-1}(0)$ for some continuous $\phi : X \to \mathbb R$, then $(X,x_0)$ is well-pointed.
For example, if $X$ is metrizable, then the zero-set condition is always satisfied.
Proof. In Showing continuity of partially defined map one can find a theorem of Arne Strøm which applies here by taking $U = X$.
Note that if $\{x_0\}$ is a closed subset of $X$, then a necessary condition for $(X,x_0)$ being well-pointed is that $\{x_0\}$ is a zero-set in $X$. See again the theorem of Arne Strøm.
Let us come back to $CH$. One can show that the only $x_0 \in CH$ for which $(CH,x_0)$ is not well-pointed are those of the form $x_0 = [\mathcal O,t]$ with $t < 1$, where $\mathcal O \in H$ is the point common to all circles. The proof is an easy consequence of Strøm's theorem.