It means “At the very beginning of the day“. It is a popular expression in The United States.
If you say I’ll do it first thing in the morning it means that you’ll do it when you get up, before you do other things like go to work, etc.
e.g,: First thing in the morning I’ll go shopping. (As soon as I wake up and dress, I’ll do this thing)
For a shorter option, you can just say “First thing.”
As in, “I’ll call mom first thing.”
That you will do it tomorrow, or in the morning is implied.
Although, saying “first thing” also seems to imply that you have a list of things to do and you’re not going to get around to it quite as soon as “First thing in the morning.”
It just means that it’s one of your top priorities of things to do.