note: This is a historical answer, and explains how to find tiny asteroids close by. The OP has clarified they want a profitably mineable-sized asteroid so there is room for more answers.
...specific asteroids and/or their asteroid orbit type...
As discussed in Have there been any documented mini-moons since 2006 RH120?, "mini-moons" are predicted to regularly enter temporary chaotic orbits within the Earth-Moon system for months to years, and for meter sizes and below there should usually be one available, if you can find it! (see below). From the last half of the abstract of the 2012 preprint The population of natural Earth satellites:
At any given time there should be at least one NES of 1-meter diameter orbiting the Earth. The average temporarily-captured orbiter (TCO; an object that makes at least one revolution around the Earth in a co-rotating coordinate system) completes (2.88 ± 0.82) rev around the Earth during a capture event that lasts (286± 18) d. We find a small preference for capture events starting in either January or July. Our results are consistent with the single known natural TCO, 2006 RH120, a few meter diameter object that was captured for about a year starting in June 2006. We estimate that about 0.1% of all meteors impacting the Earth were TCOs.
For planning and launching such a mission you do not even need a significantly positive geocentric $C_3$; your mission takes place in high Earth orbit though you will need some more delta-v for maneuvering into the asteroid's chaotic orbit, coming along side it, and station-keeping against differential solar pressure and tidal forces from Earth and Moon.
Note: you will need very tiny, precision thrusters for these maneuvers as your velocity adjustments may be measured in microns per second rather than millimeters per second!
You may also want to scrap those large, flat solar panels which in this case will feel like solar sails, and either just cover your surface with them (allow for ways to radiate heat as well!) or use an RTG.
It is worth noting that in chaotic three-body motion (Earth + Moon + negligible/essentially massless object) you can't really say for sure if the energy is positive or negative; if the orbit is bound or unbound. If you know state vectors exactly you and predict if it has a chance of escape or not and roughly when. For more on that see @MarkAdler's answer to Was the Apollo spacecraft always gravitationally bound to the Earth-Moon system? which begins:
Hmm. This question is trickier than it sounds. To know whether a body in a three-body system is always gravitationally "bound", you have to look into the future.
For a model of what the mission might be like, look at missions that insert into 3-body halo or Lissajous orbits, like SOHO, DSCOVR, JWST... "and the rest."
You will likely need a little more delta-v reserve to catch up to and match the particular mini-moon's trajectory exactly since with a mass of only a few ton's entering orbit about it will not be practical; solar pressure is probably the main problem here.
You will also need a lot of fancy thermal infrared imaging and radar work from Earth, from other spacecraft and from your spacecraft to nail down the asteroid's exact position and velocity because:
- Chaotic motion means that small errors tend to amplify exponentially with time; if you're off by a few tens of meters in position or even a few millimeters per second in velocity, a month later it could be surprisingly far away from where you predict. SOHO's doubling time was two weeks and it was in a nice, well-behaved halo orbit! See also Roberts 2002 The SOHO Mission L1 Halo Orbit Recovery From the Attitude Control Anomalies of 1998.
- The visible light reflected by a 1 meter astroid is pretty tiny, and the amount of radar power you can illuminate 1 square meter at 500,000 km is tiny as well (remember that your return signal scales as $1/r^4$), but this low-albedo, high-emissivity body is constantly radiating hundreds of watts of thermal infrared being heated at by the Sun at 1 AU, against a pretty black background.
Happy hunting, there's probably one out there right now!
Here is an example of one theoretical orbit from here or here: