Do I did my best, I made the best of it and I made the most of it have the same meaning?
No.
I did my best..... I could not have done any better, no matter how hard I tried.
I made the best of it........... Things were not perfect, but you put up with the faults and still got some satisfaction,
I made the most of it............. Things were going your way so you rode your luck and did very well.
" I know I lost my tennis match, but I did my best."
"The weather was very windy and blew the ball about, but I made the best of it."
"My opponent got very tired, so I made the most of it."
buggles (not a teacher)