Jesus used the expression 'my Father' because he knew who had fathered him. As the virgin birth account describes - God fathered Jesus by His spirit. It's already been explained (by Tony) that 'Father' was not a new term.
What is interesting though is that Jesus, a man (as every NT verse describing Jesus depicts [A]) could do nothing except the Father gave him - words, life (twice), judgement, glory etc[B]
That the Jews didn't have a clue what was going on - what Jesus was doing or saying, or even who he was, is testament to the erroneous conclusion they drew from Jesus calling God his Father - that he was 'God' also! Or, somehow 'equal' by the same faulty reasoning.
A statement Jesus, or his disciples/apostles never made. (to think that Thomas is an exception, is to misunderstand his exclamation and to ignore the rest of scripture)
From our vantage of voluminous scripture, we can clearly see that the Jews were mistaken.
They desperately trumped up all manner of charges against him - ALL were false!
What does he mean by 'Father'? It is an affirmation of his 'righteous' state - being without sin or the propensity for sin because he was uncorrupted (not being born from Adam's line)
As he had expressed, 'you are of your father, the devil...' Drawing a clear difference between who he was OF and who they were OF. (John 8:44)
Jesus having similar capacity in his roles with his God like judgement or raising the dead etc, does not of itself make Jesus equal with God - roles that the Father gave him. Being uncorrupted and without sin makes him equal in some respects but he is still mortal and about to die, he still has another will to the Father (John 6:38, Luke 22:42) These attributes make him NOT God and certainly not equal!
As an image of God (Col 1:15, 2 Cor 4:4), logic tells us that he isn't God, but a fine expression of who God is - love, grace, mercy, etc. In the terms expressed consistently through the bible as a mortal man of flesh he is certainly not equal to God. Calling God his Father is simply a fact of him being begotten as a son and the Jews despised him for it - as he exposed their wickedness!
[A] Acts 17:31 ‘God will judge the world through a man He has appointed
Rom 5:15 ‘the grace of one man Jesus’ ,
Acts 2:22 ‘a man approved of God’
1 Tim 2:5 ‘One God, one mediator, the man Jesus’
John 8:40 ‘a man who has told you the truth’
[B] John 5:26 ‘The Father gave the son to have life in himself’
John 17:7-8 ‘the words which God gave Jesus’
John 17:5 ‘Now, Father, glorify me together with yourself’
Acts 2:24 But God raised Him up again (this fact, that God or the Father raised Jesus, is repeated well over 30 times in NT)
John 12:49 ‘I have not spoken on My own, but the Father who sent Me has commanded Me what to say and how to say it’,
John 5:27 and He gave Him authority to execute judgment