I've looked around for a good soldering station and the best thing that I could find for the price/performance is the TS100. All the other budget (sub $100) didn't have a direct heating element. In total, I probably spent about $100 to get a stand and other things needed for soldering.
I'd second the TS100. It's ~$40 (given you can find a power supply for free, laptop bricks work fine. I'm using mini photo printer power supply which I found on the street + $1 plug + $3 worth of heat resistant silicone cable).
Warms up in seconds, open source and customizable firmware, takes very little space (kind of important to me since I don't have a dedicated shed/garage for the hobby). What's not to love about it for a beginner?
Apart from reading the buttons that control the device and writing the display, a digital iron needs firmware to control the feedback loop that maintains the temperature. Digital irons generally have PID based temperature regulation, unlike the crappy analog ones that have open loop systems (some analog irons have temperature regulation, e.g. Weller's old magnetic tips, but the crappy $5 irons do not). This requires a controller to supervise, but also means the iron heats up much faster and maintains that temperature while soldering.
Other features like a temperature graph or adjusting calibration and control parameters are handy additions when you have a nice display like in the TS100. Realistically though, people aren't really writing their own TS100 firmware with the exception of a few tinkerers. What it comes with is good enough.
Can you say where you can find a TS100 for ~$40? The cheapest I'm seeing is $50-55.
Also, are there different versions you need to worry about or are they all the same? I assume you only want a "MINI TS100" vs. some of the other knock off brands.
I use a Weller W61C (80-90€), which is a soldering iron (not station) that regulates its temparature using magnetic tips. Bit of an oddball, but it works fine even for soldering most SMD stuff, and it's just delightfully simple. Just plug it in and start soldering. Takes up basically no space too, since it's just a soldering iron with no station.
For really fiddly SMD items I have a cheap hot air rework station, but I don't need it very often - mostly when I have to desolder something SMD or solder on an IC with a thermal pad on the bottom.
I could not recommend the DSO203 in good conscience, it has almost no isolation of the input channels which makes it downright dangerous.
It's nice if you only do audio or low voltage stuff but the first time you're going to point your probe at something a bit more beefy and you will let the magic smoke out (good case) or worse.
