Assuming the DM is on board with the premise, are there resources that would provide some guidelines on how a PC may create a spell, and mechanics that should be considered for balancing and fairness purposes?
How can I let my players develop new spells without breaking the game?
Ideally these resources wouldn't necessarily be limited to the 5e mechanics, if previously published materials can be translated appropriately or be used as a base model.
If a spell is so good that a caster would want to use it all the time, it might be too powerful for its level.
A long duration or large area can make up for a lesser effect, depending on the spell.
Avoid spells that have very limited use, such as one that works only against good dragons. Though such a spell could exist in the world, few characters will bother to learn or prepare it unless they know in advance that doing so will be worthwhile.
Make sure the spell fits with the identity of the class. Wizards and sorcerers don't typically have access to healing spells, for example, and adding a healing spell to the wizard class list would step on the cleric's turf.
There is also a "helpful" chart for average damage by spell level. I'm going to include it, but I personally feel that the chart is not consistent with actual averages for listed spells. Please glance at the chart, and then continue on to my next point below.
\begin{array}{lll} \rlap{\textbf{Spell Damage}} \\ \textbf{Spell Level} & \textbf{One Target} & \textbf{Multiple Targets} \\ \hline \text{Cantrip} & \text{1d10} & \text{1d6} \\ \text{1st} & \text{2d10} & \text{2d6} \\ \text{2nd} & \text{3d10} & \text{4d6} \\ \text{3rd} & \text{5d10} & \text{6d6} \\ \text{4th} & \text{6d10} & \text{7d6} \\ \text{5th} & \text{8d10} & \text{8d6} \\ \text{6th} & \text{10d10} & \text{11d6} \\ \text{7th} & \text{11d10} & \text{12d6} \\ \text{8th} & \text{12d10} & \text{13d6} \\ \text{9th} & \text{15d10} & \text{14d6} \\ \end{array}
If one assumes that the values are averages, with saving throws accounted for, then it starts to make more sense.
The game also encourages you to reflavor spells (,monsters, items, etc) as you see fit. Changing fire to ice or lightning to acid shouldn't impact too many things, but be careful with the rarely resisted force and radiant damages.
We've used "Ball of Ice" instead of Fireball for one of the wizards in my game, to no harm (to the game; the monsters were quite harmed), but I would be extremely leery of using "Force Ball".
It's entirely possible that you accidentally create a spell that is either a complete dud, or horrendously overpowered. Sometimes something looks good on paper, but not so much in practice.
It's important to communicate with any players that are in the business of creating spells that you might have to tweak things as you go along. We all had a good laugh that one time you dealt 200 unpreventable damage in one turn, but we're going to have to adjust these dials a bit.
In short, follow the spirit of the general mechanism behind all of D&D:
The DM describes the environment.
The players describe what they want to do.
The DM narrates the results of the adventurers’ actions.(headings from the Player's Handbook: How To Play)
So, have the player describe what they want the spell to do, in in-world terms, and then, as the DM, determine a mechanical representation for it.
The more usual way to handle this is to allow the player to describe the spell's effects and select the level that they want the new spell to be, and as DM, decide on the damage dice, saving throws, etc. based on what is appropriate for that level.
In some cases, the player may ask for the spell to do more damage (or last longer, or some other power-related change), in which case you can offer to allow that change if the spell's level changes (as deemed appropriate by the DM) to represent its new power level.
Basically, the player describes the effect, and can then choose either the damage range or the spell level, but not both.
If, at a later date, the DM feels that the spell is not balanced compared to similar spells at similar levels (either too strong, or too weak) then a similar mechanism can be used to correct it: simply ask the player if they'd like to keep the damage and change the spell level to match, or keep the spell at its current level and adjust the damage to be more suitable. This also applies if it's the duration or some other part of the spell that needs adjusting.
When designing the mechanics of a spell, look at existing ones first. If you can change the descriptive text and keep the numbers the same, then do that. If not, then re-using phrases and sentences from other spells is often a good idea - consistency makes it easier to remember, using tried and tested wording makes it less likely that you'll introduce unintended behaviour. Remember to check the Errata and Sage Advice to see if the spells you're copying from have been amended since publication.
If you aren't modifying an existing spell (or you are, but you're changing its level), use the spell damage table on p283 of the Dungeon Master's Guide as a starting point, but don't be afraid to deviate from it as needed. Looking at the existing spells, you'll see that almost all of the cantrips that deal damage deal less than the table suggests they should, while some spells deal more than recommended (famously, fireball deals 8d6 at a level where that table suggests 6d6). A lot of these deviations were introduced during playtesting. A few rules of thumb:
Have them tell you want they want it to do, and create the specifics yourself. If they're not completely happy with the result, their character can do more research to adjust the spell.
Creating new spells needs to be done cooperatively between the player and GM, so that the player is happy with the result, and the GM is satisfied with the balance.
I suggest your ask the character what they're going for with the spell, then create a 'prototype' of the spell. Perhaps err on the side of caution in terms of balance. If they like it, great. If not, they can do more research to adjust it. If it feels weak, you can adjust it between sessions, and justify this as the caster further refining their new spell, without having to expend more in-game downtime on research (as requiring further investment of resources might feel unfair).
Piggybacking on Maaark's answer here, what I'd add is that the difference between inventing a spell and merely executing a known one ought to be a bit like the difference between computer programming and simply running a program, or perhaps the difference between following a known recipe vs. developing yourself a new one.
In other words, it should take lots and lots of time, use up a lot of components, there should be interesting failures along the way, and in the end the player may have to accept trade-offs they didn't anticipate. For instance, it might take months to figure out how to do it in a way that doesn't have a 20% chance of fizzling every round, and doesn't turn the caster's face bright blue for a week.
Think Edison working on the light-bulb. People who perfect spells generally don't waste their time traipsing about the countryside engaging in altercations with the local fauna. In fact, if they don't have a lifetime to tinker on it (and a legion of flunkies like Edison did), they might never quite get all the kinks out.
