In doing some research on this subject, this is what I found. According to the ISBE
She [Abhishag], had an intimate knowledge of the condition of David, and was present at the interview of Bathsheba with David which resulted in the placing of Solomon on the throne. If that act had been questioned she would have been a most important witness. By reason of this and of her personal charms, she might become a strong helper to any rival of Solomon who should intrigue to supplant him.
This could have made Solomon suspect to a coup against his throne
Easton's also states on that line of thought:
After David's death Adonijah persuaded Bathsheba, Solomon's mother, to entreat the king to permit him to marry Abishag. Solomon suspected in this request an aspiration to the throne, and therefore caused him to be put to death
It is also considered, that even though she was not officially one of David's wives, she was Davids wife just as any concubine would be, to this point Nave's states:
wife of David, 1 Kin. 1:1-4; 2:13-25.
Which would have been a capitol crime in Israel, for a son of David to lay with one of his wives.
Lev 20:11 If a man has sexual intercourse with his father’s wife, he has exposed his father’s nakedness. Both of them must be put to death; their blood guilt is on themselves.
Deu 27:20 ‘Cursed is the one who has sexual relations with his father’s former wife, for he dishonors his father.’ Then all the people will say, ‘Amen!’