In all Terminator movies John Connor's mother is Sarah Connor while his father is Kyle Reese.
Does anyone know why John uses his mother's surname "Connor" when his father's surname was "Reese"?
If you remember at the end of the first Terminator movie, Sarah Connor was recording tapes of everything that happened for John to listen to, and one of the important details was she told him he had to send Reese no questions asked-- which means he had no idea that Reese was his father at least until he could probably deduce from the tapes, even though it's never explicitly said.
I postulate that John grew up not knowing who his daddy is. And even if you find out when you're 20 what are you going to do change your last name?
Because John is a bastard child of a one-night stand that typically ends with the father dying at the hands of a killer android and the unwed mother raising him alone 9 months later. A child of that situation rarely gets the father's name, in movies or real life. A bastard being:
1. a person born of unmarried parents; an illegitimate child.
Slightly insulting depending on the usage, but that's the meaning. In some laws, a bastard doesn't have the right to inherit property or the surname of the father.
John also sees his mother as strong, so would never drop her name from his. He also never met his father as a child, and wouldn't want the name of the man who "abandoned" him (child psychology).
Then you have the time travel aspect. Kyle told Sarah about John Connor, Sarah names him Connor, John grows up and saves Kyle as Connor, Kyle travels back in time and tells Sarah about John Connor. Rinse, Lather, Repeat. It's a predestination paradox.
Out of universe, the reason is simple. That was the plot twist. If John's name was Reese, it would ruin the surprise.
Let's take a look from the very practical side - a baby is born, is given a name and issued a birth certificate.
The mother isn't married, so there isn't an automatic assumption of paternity. In order to list someone as a father of the child, you need either a signed consent from the father, or a court decision establishing paternity. Kyle Reese cannot give consent, since he's dead. A court decision establishing paternity requires identifying the father - and that's not really possible in the case of a time traveller like Kyle Reese; legally, he doesn't exist; the only suggestion that he's actually called Kyle Reese is her own memories. You can't list someone as a father just because you want to and claim that it's true. You're simply not getting a court decision to establish paternity with an unidentified (possibly made up) person without hard evidence. Even more, you're definitely not getting a paternity judgement if you don't want to, and it's quite likely that Sarah does not.
So, from the legal perspective, John Connor's mother is Sarah Connor and the father is not listed. The only surname normally available is Connor.
There are 3 easy reasons why Sarah would have named the child Connor and not Reese.
First, when being asked who the father was it is easier not mentioning that he was a solider from the future in a army led by your son, and generally keeping all information at a minimum.
Second, evil robots from the future are trying to kill Sarah's son by killing her, so she would know that would make any husband a potential target and if the robots found out who he was and that he was from the future they would realize they could achieve victory by simply not building a time machine in the first place.
Lastly, if Kyle Reese was a favored solider under John Reese, and John gives him a picture of his mom as soon as anyone hears about a time machine, everyone would figure out the plot.