Going mostly on a level of intellect and choices:
Jobs/classes - the user can assign a job/class the character is skilled at.
Morality - the user can assign the character a morality on a scale from good to evil.
Introvert/extrovert - the user can make the character shy or talkative.

Oh attributes! Good idea. I'll also add:

Name generator - Could be based on sounds that resemble the species, or combination of words that resemble the attributes of the current configuration.

Oh attributes! Good idea. I'll also add:

Name generator - Could be based on sounds that resemble the species, or combination of words that resemble the attributes of the current configuration.

Use that but also give the players the option of naming the characters, for a personalized feel :).

Gender agnostic - There won't be a gender selector, just a menu of species, and the user is free to modify the character to look as preferred.

Depending on the mechanics of the game wherein the character is used, you may get into trouble when doing so. Thinking of text (descriptions, dialog, thoughts, ...) in e.g. RPGs, you typically have the need for personal pronouns which are, in many languages, gender specific. Just repeating the proper name again and again isn't nice to read. I personally would not drop this property in general (assuming that the biology of the supported species isn't genderless, of course ;) ).

you typically have the need for personal pronouns which are, in many languages, gender specific

Very good point. A gender preference should be available for conversations only.

Gender agnostic - There won't be a gender selector, just a menu of species, and the user is free to modify the character to look as preferred.

Depending on the mechanics of the game wherein the character is used, you may get into trouble when doing so. Thinking of text (descriptions, dialog, thoughts, ...) in e.g. RPGs, you typically have the need for personal pronouns which are, in many languages, gender specific. Just repeating the proper name again and again isn't nice to read. I personally would not drop this property in general (assuming that the biology of the supported species isn't genderless, of course ;) ).

Are you saying you can't refer to them as an "it"?

Gender agnostic - There won't be a gender selector, just a menu of species, and the user is free to modify the character to look as preferred.

Depending on the mechanics of the game wherein the character is used, you may get into trouble when doing so. Thinking of text (descriptions, dialog, thoughts, ...) in e.g. RPGs, you typically have the need for personal pronouns which are, in many languages, gender specific. Just repeating the proper name again and again isn't nice to read. I personally would not drop this property in general (assuming that the biology of the supported species isn't genderless, of course ;) ).

Are you saying you can't refer to them as an "it"?

Not for a person, no. That would be awkward at best, dehumanizing and insulting at worst. You could have a gender-neutral pronoun as an option though (sie/hir and ey/em are two of the more widely used ones).

Gender agnostic - There won't be a gender selector, just a menu of species, and the user is free to modify the character to look as preferred.

Depending on the mechanics of the game wherein the character is used, you may get into trouble when doing so. Thinking of text (descriptions, dialog, thoughts, ...) in e.g. RPGs, you typically have the need for personal pronouns which are, in many languages, gender specific. Just repeating the proper name again and again isn't nice to read. I personally would not drop this property in general (assuming that the biology of the supported species isn't genderless, of course ;) ).

Are you saying you can't refer to them as an "it"?

Not for a person, no. That would be awkward at best, dehumanizing and insulting at worst. You could have a gender-neutral pronoun as an option though (sie/hir and ey/em are two of the more widely used ones).

I didn't know that. To weigh in, I've heard of people on the internet not wanting to disclose their genders and when asked whether to refer to them as "he" or "she", they just say, refer to me as "it".

That brings an off-topic question to my mind, how do you determine the gender of a word in English? In other languages, i.e. Spanish, the last vowel of a word defines its gender, with only a few exceptions. In English would you say that a "ship" is a she or he? English is not my native language as you might realize so I'm curious