I'd like to suggest that we need not wait for an appropriate asteroid to come within range of Earth and plan human missions around just those opportunities - we can engineer a perfect close approach.
Next year, at the start of June, the asteroid 2009 BD will pass within the orbit of the Moon. Even though it is not a rare occurrence, few opportunities such as this have been identified, and realistically, no-one is going to be ready to send a human mission to an asteroid for years to come.
It seems a shame to let this close approach go to waste, with just 40m/s of tweaking, the orbit of 2009 BD could be changed to this:
The blue part indicates the part of the orbit which is within 20 lunar distances of Earth: over 965 days in the next 6 years. The delta-v required to change the orbit of this asteroid could be delivered by a single impulse, and the most advanced technology available to do that is to simply run into it.
Calculating how to change an asteroid's orbit has been an educational exercise which would not have been possible without the JPL Horizons system and the SPICE toolkit.