Noone really knows why Julius Caesar chose Gaius Octavius,* or what his plans were. In the words of Mary Beard, "Why Caesar favoured this young man will always be a mystery." That said, I think we can have a go at figuring it out.

The only explicit indication I can find at the moment in a primary source is from Suetonius, which is a bad start! If you read a bit of Suetonius he starts to feel like more of a tabloid than a newspaper, more an entertainer and rumour-mongerer than serious historian. But nonetheless, he provides an interesting anecdote. Young Octavius, already strongly attached to Caesar, had received military decorations in Caesar's African triumph at the age of 16, even though he had been too young to fight. Caesar then departed to fight Pompey's sons in Spain (c. 45 BC), and desperate not to be left behind even though he was very ill, Octavius undertook a difficult and dangerous trip to reach Caesar with only a small party, which allegedly "delighted Caesar, who soon formed a high estimate of Augustus' character" (Suet. Aug. 8). However, Suetonius being Suetonius, this may not have actually happened.

The historicity of this anecdote aside, Eck also claims that there is no evidence Julius Caesar had "made specific plans for his great-nephew." However, he does make the point that Caesar, like all elite Romans, thought in dynastic terms and wanted an heir so that his rank and privileges could be passed on in his family. Caesar had no children after the death of his daughter, and his closest male relatives were three great-nephews, of which one was Octavius.

Considering Caesar was engaged in dangerous business in fighting a civil war, he probably wanted to be sure provisions existed for his name and achievements to be carried on by an heir if he did meet an untimely death**. I don't know why he didn't have children, but as he appears to have taken Octavius under his wing several years before his death, he appears to have had him in mind for some time as a potential heir, even if as a stop-gap measure until he could have his own son.

Octavius' later career as Augustus clearly demonstrates his organisational, intellectual and leadership qualities, some spark of which Caesar must have seen. Augustus' lack of real ability as a general is forgivable in that he was aware of it, and wisely appointed able men such as Agrippa to take charge of tactics. Julius Caesar was a highly effective and intelligent man, and he must have seen some of these same qualities in Octavius, even if they took a different form. His lack of an heir and the turbulence around him after he marched on Rome made it necessary for him to appoint one, and so in the circumstances Octavius was probably his best option. His choice was certainly validated over time: Octavius, later Augustus, ended the civil strife that had plagued Rome since the Gracchi and found a serviceable answer the problem of military intervention in politics started by Marius and, fatally for the Republic, confirmed by Pompey. I think (and Eck agrees with me here) that it was probably fortunate for Rome that Augustus was so capable, and so successful, and they had Caesar to thank for that.

Sources:

Suetonius, Life of Augustus, 8.

Mary Beard, SPQR, 340.

Werner Eck, Age of Augustus, 7-8.

*Octavian apparently just meant something like "ex-Octavian", it was never a name he used for himself. He would have called himself Caesar.

**Eck however claims that Caesar probably expected to have more time to sort his will out, provide more specific provisions, and set out a more stable plan for succession.