"Now Adonijah, the son of Haggith, went to Bathsheba, Solomon's mother. Bathsheba asked him, 'Do you come peacefully?' He answered, 'Yes, peacefully.' Then he added, 'I have something to say to you.' 'You may say it,' she replied. 'As you know,' he said, 'the kingdom was mine. All Israel looked to me as their king. But things changed, and the kingdom has gone to my brother; for it has come to him from the Lord. Now I have one request to make of you. Do not refuse me.' 'You may make it,' she said. So he continued, 'Please ask King Solomon - he will not refuse you - to give me Abishag the Shunammite as my wife.' 'Very well,' Bathsheba replied, 'I will speak to the king for you.'" NIV translation
Upon David's death, Adonijah went to the king's mother and it seems like he had accepted the fact that Solomon was the king and even seemingly acknowledged that it was God's will. He made what appears to be a simple request and, not asking for much, Bathsheba agreed to ask her son on Adonijah's behalf. We recall, from chapter one, that Abishag was the girl that was to keep King David warm in his old age and Adonijah simply wanted her as a sort of inheritance.
"When Bathsheba went to King Solomon to speak to him for Adonijah, the king stood up to meet her, bowed down to her and sat down on his throne. He had a throne brought for the king's mother, and she sat down at his right hand. 'I have one small request to make of you,' she said. 'Do not refuse me.' The king replied, ' Make it, my mother; I will not refuse you.'" NIV translation
What a beautiful picture we see here of the proper respect a son should have for his mother. We see this in the fact that he stood up, bowed down to her, & had a throne brought for her.
"So she said, 'Let Abishag the Shunammite be given in marriage to your brother Adonijah.' King Solomon answered his mother, 'Why do you request Abishag the Shunammite for Adonijah? You might as well request the kingdom for him - after all, he is my older brother - yes, for him and for Abiathar the priest and Joab son of Zeruiah!'" NIV translation
Solomon became angry at this insult but it is clear that Bathsheba had not known what she was asking. In this request, Solomon sees the fact that Adonijah had not given up his hopes of replacing him as king.