King James Version O LORD, truly I am thy servant; I am thy servant, and the son of thine handmaid: thou hast loosed my bonds. [The ASV, RSV, NIV, NKJV and ESV are very similar.]Reading the verse in several versions of different styles helps to bring out the meaning, I think. I am taking it that the psalmist is emphasising that he was born into a family which serves God, as Timothy was. [See 2 Timothy 1:5.]
New Revised Standard Version Psalm 116:16 O LORD, I am your servant; I am your servant, the child of your serving girl. You have loosed my bonds.
New Living Translation, 2nd edition O LORD, I am your servant; yes, I am your servant, born into your household; you have freed me from my chains.
Today's New International Version Truly I am your servant, LORD; I am your servant, as was my mother before me; you have loosed my bonds of affliction.
Contemporary English Version I worship you, LORD, just as my mother did, and you have rescued me from the chains of death.
Good News Translation I am your servant, LORD; I serve you just as my mother did. You have saved me from death.
New Jerusalem Bible I beg you, Yahweh! I am your servant, I am your servant and my mother was your servant; you have undone my fetters.
But I like the way the freer translations bring out the thought that he is serving God, as his mother served him. Not everyone can say this, but if we can't, we can aspire to our own children one day saying it of us, as we faithfully, though imperfectly seek to serve our God and Saviour, Jesus Christ.