Mustafa مُصطَفى and istafa اِصْطَفَى in the Quran
...Allah has chosen for you this religion... 2:132
Indeed, Allah chose Adam ... 3:33
..."Praise be to Allah , and peace upon His servants whom He has chosen. ... 27:59
Has He chosen daughters over sons? 37:153
...He could have chosen from what He creates whatever He willed. ... 39:4
...among the chosen and outstanding. 38:47
Allah used the verb istafa "choose" in first place when he spoke about His religion, His Messengers (namely Adam, Noah and the family of Abraham and the family of 'Imran) and creations he used the word Mustafa when quoting His Messengers -in the Quran explicitly- Abraham, Isaac and Jacob. So all of these are choices for the whole mankind.
Mujtaba مجتبى, ijtaba اجتبى and jaba جبَى and its meaning in the context of the Quran
jaba has the meaning collect and can be found once in the Quran (28:57).
ijtaba has different meanings:
one of them may be fabricate:
... "Why have you not contrived/chosen it?" ... 7:203
An other is to choose (for oneself):
... But [instead], Allah chooses of His messengers ... 3:179
And thus will your Lord choose you and teach you ... 12:6
... Allah chooses for Himself whom He wills ... 42:13
In the context of Messengers both the name Mustafa and Mujtaba are valid, but Mustafa has a more general meaning, as it means chosen (in general) not chosen for a specific person/tribe/nation etc. so in this context Mujtaba in relation to Messengers could be considered as contradictory or a more or less "bad choice" as it isn't exact enough and allows a misleading nuance in the meaning.
Note tat both "names" in Arabic are attributes so they can be adapted to some extent or with a restriction to anybody whom could be described this way as if you said "the chosen among scientists ... ". But as said in my comment calling somebody by an attribute needs some basic knowledge in Arabic language beside a reason for calling a person by it. So to call al-Haan or al-Hussain ibn 'Ali al-Mujtaba needs some reason! While calling Muhammad al-Mustafa has some background, al-Qadi 'Iyad said in his shifa:
ويحتمل أن "المصطفى" : كان من أوصافه ، لا من أسمائه التي تجري مجرى الأعلام .
It is possible that al-Mustafa was one of his (the Prophets) attributes, not a name of the kind a person is denominated. (My own translation take it carefully)
See also this fatwa on the names of the Prophet.