I plan to make my character an astral self monk and plan to later on take rogue and possibly fighter for the psionic subclasses. For roleplay/backstory purposes I asked the DM if I could have vulnerability to psychic damage.
They said yes but want me to gain a resistance to another damage type. I had the idea of gaining the ability to cast absorb elements at first level once per short rest.
Would this be balanced, and if not what is the most balanced choice of resistance that I can make?
Detect Balance is a tool that is normally used to evaluate new races, and that provides point values for various game features like vulnerabilities or free spell use. It's not perfect, but the point values themselves are pretty well designed.
According to this, a vulnerability for a rare damage type like psychic is worth a negative 8 points, while a daily first level spell use is worth 3 points and a common resistance like poison or fire is worth 4. That means, on the surface, it could be about balanced to get Absorb Elements once per short rest to offset psychic vulnerability -- two short rests per day are pretty typical, which would give you three uses, or about 9 points worth. Using resistances, fire and poison together would together be worth 8 points, and balance it out numerically, too.
However, the value of this of course will entirely depend on how many dangers you will face that deal psychic damage. If there are no mind flayers and so on, this trade could be a big advantage.
The reason why vulnerabilities have such a large negative value is because they can easily lead to player death -- an amount of damage that normally would only be tough suddenly spells instant death. For that reason, even balancing it with resistances is not a good plan. It just does not play that well -- your "balancing" benefit is giving you an advantage over the other players most of the time, which can feel unfair to them. Your drawback very rarely comes to play, but when it does it's really bad. I generally would recommend to avoid taking a vulnerability for your character. None of the official races has any, either.
Your DM seems to be pretty considerate, and it sounds as if you are playing a game that is more interested in exploring characters than mechanical crunch (after all, you offered up to take the vulnerability without any benefit to make up for it). If so, it might be OK to just do as you plan, and use the vulnerability and Resist Elements, and embrace the consequences.
