Thanks for your post and congratulations on owning a very fine luger. At this point, unfortunately, there isn't a lot more definitive information that we can offer. We can offer some possible insight and suggestion, however. There are two keys elements to determining the value of this luger. One is he engraving and the other is the monogram. First, the engraving. Lots of guys brought home lugers and other firearms and then had them "personalized." This process often included having the weapon chrome plated and/or engraved. The question in this case is whether this luger was engraved before or after it came to the US? Your relative may say that he brought it to the US already engraved, but without some sort of documentary evidence, that story can't be verified. However, the style of the engraving offers some evidence. The pattern of the engraving is typical of German engraving, not American engraving. It is often referred to as an "oak leaf" pattern and was a favorite of several companies such as Eickhorn, which often included oak leaves in the background of it's icon squirrel logo. An expert in German engraving may be able to identify your example more definitively. Second, the monogram is the real key. If you can identify the monogram, that will give the pistol "provenance" which will translate into dollars. Fortunately, K and T are not among the most common initials. Who could this be? Well, the engraving doesn't include any party symbols, so your luger probably was not given to a party member in honor of some work for the party. That leaves us with the recipient probably being from the military or civic domain. My understanding of the rules of monograms is that the "K" imposed on top of the "T" means that the last name begins with a "K", but I'm not even certain of that. So, here is a list of possibilities on the military side to begin your search. These are not all "generals" as your inherited oral history suggests, but are people who might have been honored with a luger such as yours:
Kurt von Tippelskirch, Karl Freiherr von Thüngen (unlikely), Karl Alfred Thieme, Traugott Kempas, Theodor Koch, Theodor Krancke, Theodor Kretschmer, Theophil Kuhnle, Karl Tannert, Kurt Tanzer, Karl-Heinz Tesch, Karl Thiel, Kurt Thoma, Karl-Anton Thomas, Karl Thurmann, Kurt Tiesler, Kurt Tischer, Karl Torley, Karl Trautmann, Kurt Trippensee. The task now will be to put your relative, based on his military service record, in a geographical location where he might have come into contact with one of these people, the family of one of these people, or the home of one of these people. They are among some of the names that you might investigate. There are others and it will be a huge challenge to identify the recipient of this luger, but it will be worth it. Best of luck and keep us informed.