The following are the primary ways GDRI tries to determine the developer (company) of a game:
An "actual mention" is when a company name is actually mentioned in or on the game. Sometimes, a game or game package will say upfront it was developed/programmed by a certain company.
Other times, the mention is not-so-forthcoming. It might say "Produced by Company" or "Designed by Company." And other times still, the company might just be mentioned in the copyright notice. In these cases, we will infer that the company is the developer. Hopefully, there will be other evidence to back that up.
Another instance of what we would consider an actual mention is when the name of the developer is written on a wall, sign, or other object in a game. An example of this can be found in the unreleased Ninja Gaiden for the Sega Mega Drive. A sign flashing "Opus" backwards can be seen at the end of round 4-2. 
Thanks to a program developed especially for GDRI, we can perform code comparisons. This program can look at two different files and show what is shared between them. For our purposes, we can compare two ROMs for games we know to be by the same developer and see what code is shared between the two. From there, we can search for this shared data in other ROMs. If there's a significant amount of shared code in these other ROMs, there is a good possibility that these were programmed by the developer of the two games we originally compared.
Sometimes, this is not particularly helpful if there is a lot of generic code used or there are drastically different programming teams. Other times, this can be a powerful tool (especially if it can be backed up by other evidence).
The aforementioned program is not available for download yet, as it is not in a distributable state.
NOTE: GDRI does not condone software piracy.
Sometimes, GDRI is able to make contact with former staffers who are willing to help and share info with us. Look at our Interviews section to see who we have talked to so far.
If you are a former staffer with one of the developers covered on GDRI and want to help us, please contact us at smsgenny at vgrebirth dot org.
Company websites are great resources for information, especially if the company is open about the games it worked on in the past (thus eliminating a lot of legwork for us).
Failing that, a great resource is SIT Developer Table, a Japanese site that was the inspiration for GDRI. There, you'll find works lists for a multitude of game companies. If we're interested enough in a company listed, we might be inspired to do some of our own research.
Books, magazines, and other publications can also contain useful information. Check out the Further Reading section of a particular entry for a list of resources.
You never know what you will find when you open up a ROM (which is as easy as opening it in Microsoft Notepad). One might find the name of the developer, names of staffers, or other "secret" text strings not otherwise accessible by playing the game.
For example, take the following text string found in the Master System Sonic the Hedgehog 2 ROM:
MS SONIC¥THE¥HEDGEHOG.2 Ver1.00 1992/09/05 SEGA /Aspect Co.,Ltd
GDRI scours the Internet, ROMs, and other sources to find game credits. We might even play through an entire game just to get them. When we've gathered enough credits from various games, we can compare them, see what is shared between them, and hopefully come to some sort of conclusion.
As we mentioned before, credits can be found in the ROMs. Usually, these are the same credits that can be seen by finishing the game. This is not to be confused with ROM text (described above), which is typically not meant to be seen by the general public.