To translate "great by choice", we first need to find good words for "great" and "choice" (or the corresponding verb "choose").
The word magnus is a good one.
It means "great" in size but also in importance, just like the English adjective.
See the linked dictionary entry (especially section II.A) for details.
For "choice" I suggest optio.
The listed translations include "choice", "free choice", and "liberty to choose".
The Latin adjective magnus needs to be in one of the three genders.
If it refers to a male or an unspecified person, it should be masculine.
For females you need feminine and for some inanimate things neuter.
You also need to specify whether you want singular or plural.
English is ambiguous, but in Latin you have to decide.
I have chose singular masculine, so that it could be a male's motto.
For other choices, please ask.
My suggestion is optione magnus, "great by choice".
The ablative optione can be read as "by choice", "because of choice", "by means of choice", or something similar.
This two-word phrase is very concise and reminiscent of many Latin mottoes.
Let me comment on eligre est magna.
First, the verb eligere (not eligre) means "choose" in the sense of selecting from options.
This verb is good for choosing your favorite food in the menu, but I would not use it for choosing to become great.
The suggested translation is grammatically wrong, too.
If corrected to eligere est magnum, it means roughly "choosing is great".
This seems closer to "I like shopping" than what you are after.
Google Translate is woefully unreliable with Latin.
