Balaji is a name for Tiruvenkatamudaiyan Srinivasa Perumal. What is the meaning of that name and what is the reason for that name? Because of this name, a few still say that it is Murugan and Shakti, which is agonizing.

It is said that the child form of Kannan during Krishnavatara himself appears at Thirumala as Tiruvenkatamudaiyan. Balaji is the name of the child Kannan. In the North, ‘Bala’ means child. ‘Ji’ is a term of respect. Combining both terms, Kannan was called Balaji during Krishnavatara. To stress that Kannan himself has come as Tiruvenkatamudaiyan, Svami Desikan says “kaNNan aDiyinai kATTum veRpu”.
Venkatachala Mahatmyam very clearly mentions the connection between Krishnavatara and Tiruvenkatamudaiyan. It also says that Yashoda herself reincarnated, et al. Hence the divine name Balaji.
There is no connection at all to say that Tiruvenkatamudaiyan is Murugan, Shakti, etc. No one has the name Balaji.