Good question!
First, remember that just because it is a solo DM/Player adventure, it doesn't necessarily mean that the PC is alone. A single DMPC (whether it be a henchman, a loyal animal companion, or an orcish prisoner) can really come in handy, and enable the character to truly shine (a warlord has someone to use his abilities on, a rogue can more easily get his sneak attacks in, and the wizard has someone to keep the enemies at bay). I'd recommend using NPCs that fill a "brute", "Soldier", or "Artillery" role, as these roles are usually the easiest to run.
Second, get rid of the railroad. "Railroads" exist in RPGs because, with a group of multiple characters, it's hard to predict the flow of the game, and a prepared course will make it easier for your group to act. In a solo game, you should be able to react to the solo player's actions much more easily. The game will be usually be much more freeform - I've found that solo play supports Sandbox gaming very easily.
Third, try to tailor the adventure to the character in a very real way. You're on the money when you suggested using many minions for a solo wizard adventure. Carry this on for all the classes - a warlord should be able to lead a small group of soldiers; a fighter should have to hold a narrow hallway; a ranger should be sniping and using the terrain to his advantage, etc. There is a switch, here - in group play, the standard operating procedure is to build an encounter with the monsters in mind (i.e., build an environment that suits the abilities of your monsters).
In a solo game, build an environment that suits your character. If you do this right, you can throw in a fight that's much more than just a monster of equal level. You can set up encounters where the ranger is a true guerilla, and the rogue is a true assassin.
Fourth, don't run adventures with a group design parameter for a single player. IN other words, many games are built with a group in mind, and the settings reflect this. Really embrace the fact that the character is alone, and run adventures that would never work for a group. Stealth-based adventures are an obvious one (stealthing in as a long-term part of the game very rarely works in an RPG, because the non-stealth PCs are just sitting there). Weird magical rituals are another, as are dream quests, weird training quests, and duels/personal vendettas. Don't be afraid to build your adventure around the solo character, and really tie everything to him.
Fifth, remember that the game is going to run MUCH faster. Think of this - in a standard game, there are five PCs and one GM. If you run a combat of 5 monsters vs. 5 PCs, the GM gets to "Act" 5 times in the round, and each player gets to act once. So, the GM has 50% of the game time, and the players have around 10% each.
In a solo game, each round will consist of the GM moving one or two monsters, and the player moving his character. Maybe an ally will also move. In other words, the rounds are going to be short, and you'll find that one combat, even a grind, should take much less time than normal. This is a good thing, but it often means the GM has to be comfortable winging things for a long period of time, when the solo PC wanders off the edge of the map.
And they will wander - in the solo games I've run, I've learned that PCs tend to be much more independant (which, of course, only makes sense).
Sixth rule - if a PC is at zero, he's pretty much dead. Think about this one. real hard. In a standard 4e game, if you're hurt, you're out for a few rounds, until the cleric or warlord gets to you and brings you back in the fight. IN many fights I've run, a PC goes down and the rest of the group isn't really that worried. The game almost seems designed to have PCs take hits and get knocked below zero. In 4e, if that happens (whether by critical hit or just an inability to heal), the game could very well be over.
Keeping in that theme, also think that second wind is pretty much useless in a solo game, unless you're a dwarf. Why? Because you'll be spending your action to heal, and most monsters will deal enough damage to knock off those HP you just restored. the only time to use a second wind in a solo game is when you spend an action point, which is kind of lame.
Seventh point to consider: Use a smaller map. There will be much less in the way of individual PCs/NPCs in the fight, so use a smaller map to encourage the use of the terrain. If you use a large map in these small duels, much of the map will be unused. You are much better to use a narrow map with many details, because it will give the player many more gaming options.