Why does it mention specifically his bones?
Radak: They are the pillars of the body and its foundation.
Why does it mention bones speaking?
Radak #1: It is a metaphor - "Libi u'Vesari Yeranenu l'Kel Chai" (84:3). In addition to his Nefesh, also his body will thank Hashem for the salvation.
Radak #2: His limbs shook amidst intent in Tefilah and thanks; it is as if they praise.
Radak #3: From the character of the formation of the body's limbs, man praises the Creator via them; it is as if they praise 1 .
Malbim: The matter in which a miracle was done, it tells and testifies about the miracle. A miracle was done to every one of my bones and limbs - each was in special danger - so all of them testify that there is no one like Hashem.
Radak: R. Bechayei and the Rambam explained so. It is a nice Perush, but it does not fit the verse.
Who is the "Oni"?
Radak: It is David. Hashem saved him from enemies stronger than him.
Radak (from Bereishis Rabah 54:1): The Yetzer ha'Ra. Hashem saves it from the ultimate robber, the Yetzer ha'Ra.
Malbim: It is the body; "v'Oni v'Evyon mi'Gozlo" is saving his property.