I played D&D solo with my dad as the player for almost six years as a kid. We ran AD&D and 3.5 D&D. Frankly, it surprises me that more people don't play the game this way. It makes for an extremely good bonding time, it's a lot of fun, and two people with a good relationship can create some very great campaigns together.
Here are a few party recommendations. In various campaigns with my dad, I have tried all three of your options.
NPCs with Player
This is used in a lot of cRPGs. That doesn't mean you want to do it. You see, as a DM, you have a lot of things to manage in a normal game. Having to manage most of the party on top of your normal NPCs, enemies, and other miscellania is just too much. To make things worse, you are caught in an awkward situation where you're tempted to metagame really hard with the NPCs. It's hard to know whether an in-party NPC really would solve a problem or suggest a certain course of action in a given situation when you are controlling them. On a regular basis, this is something a DM learns to manage. When you constantly may have to make party decisions in every single scene and encounter, it becomes tricky.
Majority NPC parties also take away from the player's achievements. Most successes will come from NPC action (effectively DM action, in this case) and be mostly out of her control. One PC cannot specialize in everything, therefore the NPCs will fill in the gaps and sap some of the feelings of success that one has in a normal party of PCs.
Solo Player
Perhaps better than the NPCs with Player, simply because it allows the player to feels pretty successful most of the time. Unfortunately, it also sets the player up for huge stacks of failure or fudging.
For example, in a hazardous situation, you know that if you let the player die, it's all over. There is no one else. So you are very tempted to fudge the game with mercy to prevent that from happening. This can make the game feel a lot easier than it should (especially since it's supposed to be solo!)
In addition, because all versions of D&D are designed for parties to specialize and develop useful roles, a single PC can rarely rise to the occasion of all encounter types. You will have to be choosy about what kind of challenges you present the player, lest you back her into a corner.
While this method needs to be handled carefully, if you can find ways relevant to your campaign which overcome these drawbacks, a single hero can a be a rewarding experience. It also makes for a deeper connection to the story. My dad and I have fond memories of a legendary campaign led by his heroic PC. There's something special about that.
Multi-PC Player
Some people argue that D&D should be designed this way by default.
There are some great examples of this in exceptional cRPGs, such as Icewind Dale. Most players love making characters (sometimes more than playing them!) and giving them the opportunity to create several is a joyous day, indeed! This system has advantages both for new and experienced players.
New players are in a situation where they can more freely experiment and not be afraid of being stuck with a single, supremely flawed character. They can try new things, not be as afraid of PC death, and are not left out if their character is stunned/paralyzed/etc.
Experienced players are in a situation where they can try to make an optimal party. They have control of the whole group! Now they can truly test their skills, if they are more gameplay oriented. If they are more roleplay oriented, they have a cast of characters to themselves that they can create in as interesting a manner as they wish, with whatever meshed backstories they see fit.
As an aside, should your wife be uncomfortable with creating multiple characters, don't worry. I was in this exact situation with my dad when we started our first AD&D campaign. Have her create just one character, then introduce several NPCs with useful skills and abilities to her party over the course of the story. But instead of keeping them as NPCs (as the first party option we discussed), give complete control of them to her directly after a short period of having them demonstrate their personality, skills, and abilities by joining the PC. It worked seamlessly for my dad and I.
I hope you two have an awesome time playing!
