No actual national championship trophy for A&M, since the Associated Press didn't start awarding its trophy until 1941 and the Dickinson System awarded their trophy to USC as the national champions that year. No player rings or watches either, since those wouldn't be in vogue until decades later. The only actual, non-retroactive trophy A&M has for their last national championship is the Sugar Bowl trophy. The players were also awarded charms for winning the SWC, and there's that infamous tapestry that's always in the signatures of the older, more deranged TexAgs posters.
A little while back A&M had a football trophy exhibit at the George H.W. Bush Library. It feels uncharitable to call it 'pathetic', but it wasn't especially impressive. They put their 1941 Cotton Bowl trophy on display (from the season we defeated them 7-0, knocked them out of the Rose Bowl, and crushed their hopes of repeating for the national championship) along with a commemorative tapestry from their 1939 and 1940 seasons, a few photos from their 1919 and 1917 undefeated seasons, and various pins, rings and game balls.
Most of the commemorative memorabilia for A&M's 1939 national title (i.e. the things that actually say 'National Champions' on them) are things A&M created and awarded themselves much later. For example, in 1998 A&M retroactively created and awarded national championship rings to the 26 surviving players from the 1939 team.
And in 2016, when AFCA put together their short-lived 'Blue Ribbon Commission' headed up by R.C. Slocum, they awarded A&M a retroactive Coaches' Poll trophy for the 1939 season after A&M went through the process of applying for a retroactive award. The commission also awarded Okie Lite a trophy for 1945, TCU a trophy for 1938, and SMU a trophy for 1935 before AFCA presumably realized how silly the process was and stopped the project.