Yes, but it would be from the pre-determined names that [I assume] already exist. Also, if you name them something outlandish, you are probably giving up the discretion.
Despite the above discussed limitations, i still think you can get guys in there that will play a specific role, but will need to manage accordingly. An example would be a left handed DH that has 0 split vs. L, but 65+ against R with big power and eye. Or, a low inning-low left split, high right split pitcher that you can hide as an LRA behind a left handed starter. So, I think there is still a way to make it work and keep vet owners from taking too much of an advantage.
The randomness that the simmy decides are items like age, patience, temper, makeup, height, weight, etc.
This is probably all just a moot discussion because none of these suggestions will be implemented, but I think it would cool.