I understand why you might want a "stable" model, but your concept still needs to be fleshed out a little more. To collapse the variance of the prediction errors onto a single number without taking into account any other information can lead to erroneous conclusions.
Let me explain.
Let's say you are using age as your only feature in your model to predict how much your customer will spend at some retail store. The variance of spend changes with age. Eighteen year old customers don't have a lot of money, so they all spend more or less the same amount. However, 40 year old customers vary wildly in their incomes, and so those with more money will spend more. Clearly, spend is heteroskedastic (i.e. has non-constant variance).
Now, assume that you train a model and deploy it to production. In the first month after deployment, lots of 18 year olds come and buy in the store. Since variance of spends for 18 year olds is small, your model's variance in predictions is small.
But then suppose in the second month, a lot fo 40 year olds come and spend. By virtue of their spend being highly variable, your model's errors will also be highly variable! It will look as if your model has degraded (or, in your own words, become unstable) when in reality nothing has changed.
All this to say, if you are looking for measure the variance of your prediction errors, you need to condition on something. Usually, you look at the error as a function of the predictions (like I've mentioned in my comment).
So, if by "stable errors", you mean that the variance of errors is constant across predictions (i.e. the errors are homoskedastic), then makes more sense. If that is not what you mean, I think the idea needs to be fleshed out a little more for me.