1 King 2:13-18

1 Kings 2:13-18 King James Version (KJV)
13 And Adonijah the son of Haggith came to Bathsheba the mother of Solomon. And she said, Comest thou peaceably? And he said, Peaceably.
14 He said moreover, I have somewhat to say unto thee. And she said, Say on.
15 And he said, Thou knowest that the kingdom was mine, and that all Israel set their faces on me, that I should reign: howbeit the kingdom is turned about, and is become my brother's: for it was his from the Lord.
16 And now I ask one petition of thee, deny me not. And she said unto him, Say on.
17 And he said, Speak, I pray thee, unto Solomon the king, (for he will not say thee nay,) that he give me Abishag the Shunammite to wife.
18 And Bathsheba said, Well; I will speak for thee unto the king.
An American English Bible
With Old Testament based on the Greek Septuagint

13 Well later, AdoniJah (the son of AgGith) went to see BathSheba (the mother of Solomon). He bowed before her, then she asked: 'Have you come in peace?'
And he replied: 'Yes, in peace… 14 I have some business with you.'
So she said: 'Then speak.'
15 And he said: 'You know that the kingdom was mine, because all IsraEl turned to me as their king. But the kingdom was taken from me and it became my brother's, because it was given to him by Jehovah. 16 Now, I just want to ask one thing of you… and don't ignore me.'
Then BathSheba said: 'Keep talking.'
17 And he said to her: 'I beg you to speak to King Solomon, because he won't ignore you. Ask him to give AbiSag the Somanite to me, to be my woman.'
18 And BathSheba said: 'Well, I'll speak to the king for you.'

These are my thoughts, please share yours.
13 This was David’s son who was trying to take the throne away. Bathsheba may at this time have been the person you went to see to see Solomon. She asks this because he had caused trouble trying to take the throne from his father so Solomon could not get it after their father’s death. It was important that he came in peace it meant that he was no longer a problem.
14 He came to speak with her, she is Solomon’s mother but not his mother his mother is Haggith.
15-16 He understands that had he everything but because it was God’s plan for Solomon to be king, he lost the kingdom and Solomon became king, no matter what we want God’s plans will happen, we can choose to go against them or follow in His plans. After causing trouble he then wants something, how often after we mess things up do we ask God to fix things, help us.
17 He wanted to take this woman who was the one who cared for David as he grew old. It is strange that he wanted to marry her.
18 Bathsheba had no authority to grant this but only could speak to the king, he would not have wanted to go to Solomon himself because of what had happened.