I'm late to the party; I still hope you're here, though! Okay, let me start of my saying that if you were to write your sentence like so:
"My mother Martha, my father Jesse and my daughter Amy will be there."
You're leaving your readers to know that you have more than one mother, father, and daughter, and I'll exmplain why.
I'll use a good example: If you were to write, "My friend, Lia," with a comma, this makes the word "Lia" non-restrictive; therefore, it would signify that Lia is your ONLY friend. Now, if you were to write it without the comma before "Lia" as so: "My friend Lia," this makes "Lia" restrictive, showing you have more than one friend.
Now that you understand how the comma can make a difference, you need to add commas before their names; otherwise, it'll signify that you have more than one mother, father, and daughter. Since you do have a list (a list contains THREE items), and that list contains commas that describe a list item (for example, Martha describes "my mother"), it is ABSOLUTELY necessary to add the semicolons as you had it in your example. If you think the semicolon looks horrible on the eyes, you can simply put the names around parenthesis, and that'll fix your problem.
"My mother (Martha), my father (Jesse), and my daughter (Amy) will be there."
As for the comma before "and," that is optional, though I would recommend adding it to not confuse your readers. Also, keep in mind that if you're going to use the semicolons, you need one before "and." I hope this helps! :)