When planning a mission to a close approach to the asteroid Eros and to remain nearby, what are the factors that one would need to consider when choosing between ion propulsion and chemical propulsion?
This is going to depend very much on mission intent and time frame. If mission duration is in years and overall mass is small electric propulsion may work, and do so for lower overall mass. Especially if the mission on arrival requires large amounts of power so large solar cells are not wasted mass.
If aim is to get there quickly and with a large payload (eg people) generally things tend to move towards conventional thermal rockets. This is especially the case with Eros if launch time can be carefully chosen such that bulk of DV happens deep in LEO when Eros is passing relatively close to earth, allowing cryogenic propellants and maximum benefit from the Oberth effect.
For reference this handy table suggests Eros takes approximately the same amount of DV to reach as Moon or Mars, so rocket size/costs can be estimated by looking at those missions.
Noting pre edit questions about mining, returning from Eros will involve similar amounts of DV, either from a rocket or rocket+heatshield for aerobreaking and need to either be mined in situ or lifted up from earth. This would make costs broadly comparable to the moon.