It's Not Hard.
Let's examine what we mean by "balance". In a combat situation there are only three factors to consider: the average damage per round, the "nova" damage per round, and the frequency of "dead" rounds (where the character has little to contribute). In non-combat situations, there are just two things: the frequency of being able to contribute and the magnitude of the contributions.
That's all "balance" is. It is simply about the player feeling that he usually has something to contribute, and that sometimes he gets to contribute something big.
So, let's see what would happen if you simply change the 5e wizard to Vancian slots, with no other changes. We will assume that cantrips remain RAW. The "nova" damage would not change at all - a Meteor Swarm still does the same damage. The average damage per round would not change at all, either, until the wizard started running out of slots. However, as the wizard runs out of slots, he starts having "dead" rounds. In non-combat situations, the magnitude of his contributions is exactly the same, but the probability that he can contribute is significantly less, as the variety of spells he has available at any given moment is diminished.
So we have two areas to address: dead rounds in combat and fewer non-combat actions. One way to approach this would be to accept this and improve the class in other areas to compensate. This is a valid method, but it's trickier for two reasons. First, it tends to lead to love-it-or-hate-it classes such as Warlock (which, scanning this web site seems to exist almost entirely for characters to take "dips" in.) Second, whether the result is balanced depends a lot on the game style. For example, the Warlock class does little "nova" damage but rarely has a dead round. But if you DM likes to have one big battle per game instead of a sequence of battles with occasional rests, the Warlock is completely overshadowed by the "battle-casters".
The other approach is to address the specific deficiencies. @KorvinStormast suggests a variety of ways to address the "dead round" problem - magic items can address this very specifically. Another idea would be to beef up the flexibility of cantrips - adding range, adjusting damage types, increasing the number, etc.. (Note that I do not suggesting bumping up their damage - we are trying to eliminate dead rounds, not increase the damage-per-round). Note that simply adding more spell slots has an unwanted side-effect: it increases the number of times the character can go "nova" at least as much as it decreases the dead rounds. Given the choice, most players are going to load up with fireballs!
The non-combat flexibility can be addressed with similar measures. Scrolls are a particularly good mechanic for increasing the variety of actions available to a wizard - one idea would be to drastically reduce the cost of creating scrolls for the Vancian wizard, but with the limitation that only the author can use the "discount" scroll and some additional limitation regarding their use in combat.
Here's an easy idea to playtest.
The scroll idea alone may be enough. I personally like the flavor of scrolls in a Vancian class and this would be simple to implement. I would start with just this: A Vancian wizard has the same spell slots and other traits as a conventional wizard, save the Vancian mechanic. However, a Vancian wizard may scribe a scroll that only he can cast from for 10% of the normal price (per XGE pg 133). Casting from such a scroll requires extra a full Action to prepare, followed by a full Action in the next turn to cast, and no other actions (Move, Bonus, or Reaction) can occur in the interim. (This last clause limits the ability to abuse the rule for combat, while still giving it some combat utility.)
If there are any big holes in this idea, I'm sure we'll see in the comments. But this should be easy to playtest and can't be too far from balanced.