Interpretation 1:
It is a psalm of Solomon (as noted in the ESV as well as NASB, NIV).

Arguments in favour of this view:

1.    The short Hebrew word “le” as used in the psalms generally means of, as in the psalms of David, Asaph, or the sons of Korah.

2.     Even though Psalm 72 is the only psalm of Solomon in the entire book of Psalms, we know that Solomon composed many songs—more than 1,000 according to 1 Kings 4:32. A collection of eighteen songs that feature the literary form of a psalm, written by a group of Jews at the start of our Christian era, were incorporated in the first Greek translation of the Old Testament (the Septuagint) as The Psalms of Solomon. In this collection each psalm is also indicated as a psalm of Solomon.

3.    At the conclusion of Psalm 72 it does say that the prayers of David have come to an end here, and Psalm 72 is also clearly a prayer of David that he may have prayed toward the end of his life, after Solomon has received the throne. However, Solomon, as second author and editor, would have turned this prayer into a song and added it to his second bundle of Psalms (Psalm 42-72). This also reflects John Calvin’s view.

Arguments against this view:

1.    Even though the Hebrew word “le” generally means of, it often takes on the meaning to, such as: to the choirmaster (Psalms 4–6, etc.) John Calvin already knew that this Hebrew word had many different meanings (see his commentary on Psalm 72). Very often it can mean about or concerning, as in Jeremiah 23:9; Jeremiah 49:23, Jeremiah 49:7 and Jeremiah 49:28.

2.    This first Greek translation of the Old Testament, the Septuagint, already chose the word “eis” in the translation, which can mean about. The Latin translation of the Septuagint also followed that choice, as did the (Dutch) Dort Bible.

3.    Also in the header above the Song of Songs (ESV: The song of Solomon) it is—considering the content of this song—more plausible to understand it as a (protest) song about Solomon.

4.    If Psalm 72 is in fact a song of Solomon, then whose son is in view? Is it about Solomon’s son, Rehoboam? In view of his reign, this is unlikely. Would Solomon then have been singing this song about his own reign? Did he somehow anticipate the connections with the kings of Tarshish and the queen of Sheba? Was he prepared to sing so positively about himself, without any qualms? This too does not appear likely.

5.    In the New Testament David is mentioned as a prophet (Acts 2:30). In one of his last songs, he may already have been singing in a prophetic way of Solomon’s international connections. In line with this he may already have received a glimpse of this king of peace, the great King of Peace: Jesus Christ.

Interpretation 2: 
It is a psalm about or for Solomon (as found in the AKJV and other translations).

Arguments in favour of this view:

1.    The arguments against interpretation 1.

It is possible that Solomon, as the second author and editor, turned this prayer of his old father (after the latter’s death), into a song, and added it to his collection(s). However, that does not imply that we are bound to recognize this as a psalm of Solomon. As the Hebrew word can also mean for, it remains more logical and plausible to read this psalm as a psalm about Solomon.

The most plausible interpretation:

Interpretation 2