Rashi: It is for the Leviyim to say it at Chanukas ha'Bayis in the days of Shlomo. Radak - even though this Mizmor does not mention the Bayis, it mentions pardon of his Pesha'im. His enemies thought that kingship will not last for him and his sons, due to that sin [of Bas Sheva]. Even though a Navi said that Shlomo will rule and build the Bayis, they did not believe that a son born from that woman will be king and build the Mikdash, a place of pardon. When Adoniyahu tried to rule by himself, they thought that Shlomo will not rule. When Shlomo became king, and all Adoniyahu's supporters fled, they saw that Hashem wanted Shlomo to rule. They inferred that Hashem totally forgave that sin. David showed Shlomo, in front of all Yisrael, the form of the Bayis, and donated for it, and also Yisrael did.

Rashi (12, citing Pesikta Zuta): It refers to Mordechai, Esther and Haman. Haman said "Ani Amrti b'Shalvi." Esther said "Elecha Hashem Ekra" (verse 9). Mordechai and all Yisrael said "Heye Ozer Li; Hafachti Mispedi l'Machol Li" (verse 12)

Malbim: The Mizmor thanks Hashem when David was ill, and recovered. It does not discuss the Bayis at all! "Bayis" here is a metaphor; it is the body, which is the Nefesh's residence and the house of the inner person who dwells inside. The real person is the Nefesh; the physical body is merely a physical house for it to dwell in, as long as it is in the fortress -"Shochnei Vatei Chomer" (Iyov 4:19), "ba'Yom she'Yazu'u Shomerei ha'Bayis" (Koheles 12:3). When he was ill, all pillars of the house and its supports softened. After he healed, the house was built in its place and he returned to his youth.