$\require{\mhchem}$
There are almost too many ion species to list that have been used in ion trap based quantum computing or related experiments. The usual choice is one that is, when singly ionized, hydrogen-like which has convenient consequences for their laser spectroscopy: Then a strong, typically $20$ MHz wide transition lies in the UV or blue end of the laser-accessible spectrum (rather than in the vacuum-UV as it would for ions that need higher than single ionization to become hydrogen-like). Also, the spectrum remains relatively simple (if it is hydrogen-like) meaning there are a limited number of other states that may need their own laser as a repumper laser. It can be advantageous to have one optical meta-stable state that needs a repumper laser because that can be used in measurements and state preparations (or, atypically, to represent one qubit state).
Finally, you typically (but not always) want an ion that has a hyperfine structure because that allows you to use hyperfine states with only a few $\mathrm{GHz}$ energy spacing as qubit states. These states are advantageous because they have century-long decay times, meaning you have practically no decoherence simply from their spontaneous decay (but you do have decoherence from magnetic fields, to which well-chosen states have, however, no linear and only a quadratic dependence).
It is also convenient to have a low mass ion because that allows you to build an ion trap with higher motional frequencies (the ion is more strongly confined if its charge-to-mass ratio is high). High motional frequencies imply less (anomalous) heating inside the ion trap and the possibility of faster $2$-qubit gate speeds.
One of the most popular ion species is $\ce{^{171}Yb^+}$ because you have all required lasers in a spectral region (IR and visible) where you can build them with relative simplicity and there is a convenient meta-stable state of about $1\ \mathrm{Hz}$ width (and one with about $1\ \mathrm{nHz}$ width that is irrelevant), and it has a particularly simple hyperfine structure due to its nuclear spin of $1/2$. $\ce{Ca^+}$ is almost as good: If you can live without having a hyperfine structure, $\ce{^{40}Ca^+}$ has equally simple laser requirements and a relatively low mass whilst by tuning your lasers for $\ce{^{43}Ca^+}$ you gain a hyperfine structure at the expense of it being a fairly complicated one due to the nuclear spin of $7/2$. Some groups pursue $\ce{^9Be^+}$ which is cool for being so light and for only needing lasers at essentially the same wavelength, albeit a difficult one ($313\ \mathrm{nm}$). Many other ions have been used experimentally, including $\ce{Sr^+}$, $\require{\mhchem}\ce{Hg^+}$ and a good depiction of the important properties can be found at Chris Monroe's "Ion Periodic Table".
