Ok. I did a lot of research and i think i annoyed some people in the process. Ill give you what i know from a neutral point of view before bullets start flying
It all depends on what you want from your gameboy. A lot of chiptuners will prefer the DMG because it has the best bass sound. Also, they like it better because you can fit more mods into it. Also, 4 AA batteries will of course last longer than 2 AAA batteries. There was also some controversy with the MGB because it drained the battery really fast to run LSDJ. However, the new 64mb ems cartridge has fixed the issue and this is no longer a problem. Justin and Kitsch can agree.
The MGB, despite having slightly inferior sound, has the edge in being more portable. If you're on the move a lot, its a better size to carry around. Pro-sounding it can give it that extra kick it needs with the sound. The MGB is a good choice for beginners because they are only learning the program, and not doing shows. So sound is not exactly vital. On the other hand, you might prefer the DMG because you can learn on it, and then start performing without having to buy a new gameboy.
It also depends on what you plan on doing with LSDJ. Are you in to perform or do you just want a (kickass) hobby? If performing is your plan, then learning on the DMG is a better option. If you want a hobby, carrying around an MGB wherever you go is better for chiptuning on the metro or waiting in a bar for friends etc. since you dont need a carrying case for it.
MGB=pocket. DMG=original. If you didn't know already.