Is it possible to use an aftermarket controller on a MakerBot? I've got a dead MakerBot Replicator Dual and I'm not really interested in fixing it with about half the parts being proprietary, discontinued, and therefore very expensive. So, I'm wondering if it's possible to use a controller board used on some of the other RepRap machines.

I'd assume that it's all relatively the same, mostly need a means of routing/regulating power to the servos/fans/etc. based on a g-code. If that's true, then I'm also assuming the next step is ensuring the slicing engine spits out the right format for the board (.s3g, .x3g, .gcode, etc.) and that the slicing engine knows the proper bounds for the machine.

Side question, if I used an another controller board, would I be able to still run MakerWare/MakerBot Desktop? So far, I've preferred the MiracleGrue slicer, but I've used Cura on Octopi and it'd suffice.
You can put pretty much any controller in pretty much any printer, with a few noteworthy details you need to keep in mind for a Replicator 1.


Makerbots use thermocouples. Most RepRap style 3d printers use thermistors. So you would need to replace the extruder temp sensors, or choose a new controller with thermocouple support. 
Makerbots use 24v heaters and fans and PSUs. Most RepRap style 3d printers run on 12v (although this is slowly changing because 24v is better for performance and safety.) You can either pick a new controller that is known to be compatible with 24v, or replace all 24v-rated components with equivalent 12v-rated components. 
The existing heatbed thermistor circuit is probably not the same kind of circuit that your new controller will expect. Older Makerbots use a small voltage divider circuit on the HBP PCB to convert the thermistor resistance to a voltage signal, whereas RepRap style controllers typically have this circuit on the mainboard and thus need only a simple thermistor hookup. You could add a new thermistor, or modify the existing circuit, or get an entirely new heatbed. 
The existing endstops are full +5v,GND,Signal style mechanical switches with debounce capacitors and LED indicators. These should be compatible with most controller boards given proper firmware setting, but you may need to do some creative wiring if there isn't a suitable three-pin endstop port on your chosen board.
The LCD and 5-button panel will not work with other boards. You'll need to replace that or just run a remote host via USB.
Connectors may be different, of course. That's always an issue with any board swap. 
You will need to export RepRap style gcode instead of .x3g files. Makerware/Desktop can export gcode but there may be some minor differences in Mcode assignments from what most RepRap style firmware will expect. I couldn't tell you offhand whether this will work, but to be safe, it would probably be best to use a more traditional RepRap slicer (Slic3r, Cura) or Simplify3D.


So that's a healthy to-do list. At a certain point you're practically building a new printer. But there is a significantly easier approach than installing a RepRap style controller: replace the dead Makerbot Mightyboard with a compatible community-designed Sailfish board like the Azteeg x3 by Panucatt or new CloneBoard Mini by Carl Raffle. Or, if you're ok with Chinese sourcing, a FlashForge Mightyboard will be a near 100% drop-fit replacement for your Makerbot Mightyboard. Any of these options will give you fresh new electronics with minimal hardware changes in the rest of the printer. 

Either way is fine and you'll end up with a working printer. Using another Sailfish board will certainly be easier, if you don't object to staying within the Sailfish/x3g toolchain and Makerbot/Clone parts ecosystem. Makerbot does have a very bad community reputation these days, but it's important to remember that the Replicator 1 Dual is a completely open-source machine from the "pre-evil" days, and compatible spare parts are widely available from literally dozens of vendors. The Rep1-derived Makerbots and Clones collectively comprise the most popular single "flavor" of 3d printer in the world (totaling as much as perhaps a quarter of all consumer/hobbyist 3d printers sold to date) and many/most of the spare parts for Replicator 2/2x's and clones will also fit in a genuine Rep1.