Will my scar dot from a pen stabbing blue ink under the skin go away?

Probably not. Pen ink can actually act as a tattoo. The discoloration may respond to one or more laser treatments. Alternatively, and perhaps even simpler, is to excise the area as if it were a mole, but this may result in a small scar. It is best to check with your plastic surgeon.
Traumatic tattoo. It may after a year or two but it could form a permanent tattoo depending on the amount and depth of penetration.
Traumatic Tattoo. If the pigment has taken up residence in the immune cells of the skin, it may have formed a tattoo. In this case you would need a laser or surgical treatment to remove the pigment.
May be a tatoo. The ink may have actually healed in the skin, making a tiny tattoo. If it bothers you, perhaps a laser treatment can remove it. If not, it could be excised.