You need to edit your savefile. Grab notepad++ and the xml tools plugin for it. Use google to find the downloads. Go to %appdata%/StardewValley/Saves/[charactername]_[a 9 digit number] MAKE A BACKUP OF THE FILES You will need to edit both the SaveGameInfo file and the file that is named the same as the folder for that specific save. Open the files, for each use plugins>XML Tools>pretty print XML only with linebreaks. wait for it to finish, it can take a while. Search for your pets name and change it to the new name in both files. Use plugins>XML Tools>Linarise on both, again this can take a while. Save the files. Load the game.
I'd also like to rename my son, I named him Duck because my husband is Elliott and I thought it was funny at the time but now I want him to have a proper name. Where is that located?
Yeah, If you are not confident in what you are doing, one of the save editors would likely be easier. Learning how to do it yourself does have it's advantages though.
The horse isn't a Pet, but yeah, it's name (and your children's names) are in the savegame files as well. Use the 'Find' function.