To begin it's important to state the rules for the original orange Numenera corebook and the newer Numenera Discovery are verbatim on this subject. Although the page in question differs. I provide both here.
So the rules as written provide an interesting contradiction on what the players are allowed to or not allowed to do when taking mutant as a descriptor. Page 123 of the orange core rulebook or 397 of Numenera Discovery states a player must:
on the mutations tables. Which would seem pretty absolute.
However this is helpfully followed up with:
"The player and GM can work together to ensure that the resulting character is one that the player wants to play."
This indicates final say is with the GM to decide whether or not a character is viable and that a player should not be forced to play a mutant they do not want to play. There are many reasons a character with a certain mutation may be considered overpowered in any one particular game, the ability to breath underwater, or fly, or subsist through photosynthesis could introduce a new element into the game that the GM would not want to have to handle.
In your scenario it seems to me the player is looking to work with you to create a mutant that they would want to play, balance wise the key part of the mutant experience if following the rules with regards to the number of beneficial, harmful, powerfull and distinctive mutations with cosmetic mutations being entirely optional anyway. This ensures the net benefits and negatives from being a mutant compare favourably with any other descriptor in the game.
With that in mind one thing you may have overlooked and should check is that the player isn't gaming the system with knowledge shared with them about how you want to run the game unless you want them to. For example an aquatic mutant would make more sense in a campaign with more submerged elements. I would also not approve of a player using free reign to select their mutations to min-max their character.
In an aborted game I ran recently I allowed my player to choose the distinctive mutation they wanted after discussion. But I had them roll everything else. Mileage will vary between cases, some quick maths tells me there are at least 235872 different combinations of mutants in the orange core book without taking in to account cosmetic mutations or the expanded list of mutations in the "character options" book some of those combinations will be slightly more powerful than others. Use your discretion.
The mutant descriptor's randomness provides for players who enjoy discovering their character during the process. I consider it an intrinsic part of picking that descriptor as that's how the "always rolls" parts of the rules indicates to me the author intended it to be played. But there's nothing to say you can't break with it if it suits you. The mechanical impact of descriptors tends to fall away as the game goes on as they don't scale like Focuses or Types. Selecting a mutant though in a world which regularly discriminates against their kind opens up many roleplaying opportunities for good and ill.