As best I can determine in 2 Sam 7:14, אֶהְיֶה is simply the verb "to fall out, come to pass, become, be", etc. In the absence of the pronoun, it would still be translated "I will be", or I become" etc.
However, the inclusion of the pronoun, אֲנִי֙ ("I") simply adds emphasis to the personal nature of the relationship promised between God as a "Father" and the promised royal dynasty as the "Son".
This statement is "pregnant" with prophetic and Messianic overtones. Later, the "Son of God"/"Son of Man", Jesus, Messiah would be born the the royal line of David and be called the Son of God as well.
While David and his successors were earthly kings, they were to recognise that the real king of Israel was God. 1 Sam 8:7, 8, 24:6, 2 Sam 19:21, 1 Chron 28:5, 2 Chron 9:8, 13:8, Ps 5:2, 44:4. See also 1 Sam 12:14.