Okay, let's begin with some rough arithmetic.
The mean radius of the music on a 78-rpm disc is of the order of 4". The length of one revolution at that radius is around 14", so a fragment an inch across will be about a fourteenth of a revolution. 78 rpm makes that about a milli-minute <G> or a sixteenth of a second. So to begin with, quarter-sized fragments are not going to help.
If there's a fragment large enough to be useful, retrieving information will be a major adventure. For example, if the shard includes some of the center hole (for location) and has some groove in an arc of perhaps twenty degrees, it might be worth experimenting to get an idea of the final sound, hence insight into the recording. But remember that a full rotation of the disc is less than one second; it takes a lot of that rotation to get sense out of the recording.
Thanks for your comments. I realize that it's a long shot to get any useful information from these fragments. He's actually hoping to find cross-mends between the sherds to build larger pieces, but they're still in the fieldwork phase on this project and haven't analyzed what they've found yet, which also means that they might have some larger pieces and he dosen't know it yet. I know from experience that, depending on context, it could be possible to find most of the fragments of an object like this (for example, if someone had swept up the broken pieces and dumped them together into a trash pit). From an archaeological point of view, even getting a good guess at the genre would be great information. I also just ralized that it might be possible to get an approximate idea of the position of a fragment on a record without the center hole by calculating the radius from the arc of the grooves.
Thanks, Aaron