The banner in question is that of the "New Orleans Greys" 1st Company. Volunteers for the cause of Texas independence, the two companies of Greys were, according to the Handbook of Texas Online, organized at a mass meeting in a New Orleans coffee house in October 1835. The flag of the 1st Company was presented to the unit by ladies from east Texas shortly after the company arrived in Texas. According to the Handbook the flag was of blue silk, with an Eagle and Sunburst, and the inscription "FIRST COMPANY OF TEXAN VOLUNTEERS! FROM NEW-ORLEANS." The Eagle carries, in his beak, a placard with the motto "GOD & LIBERTY."
The two companies arrived in Texas early in the Revolution, both units participating in the capture of San Antonio from General Cos, but shortly thereafter meeting their ends as organized units -- the 2nd Company going down at Goliad, the 1st Company trapped at the Alamo. Seven survivors of the two companies, however, were present at San Jacinto.
On the morning that the Alamo was stormed, the banner of the 1st Company was captured by the Mexican Army, and the flag (sent to Mexico City with an accompanying cover letter by President-General Santa Anna to Mexican Secretary of War and Marine José María Tornel) is now in the possession of the National Historical Museum in Mexico City.
Various attempts have been made to reclaim the flag, all unsuccessful. Now we have Representative Zerwas's bill, which, boiled down, amounts to a request to Mexico that it be nice and give us the flag back. The text of the bill directs the Governor to "negotiate" the flag's return from Mexico, via "purchase or lease."
I'm as proud a Texan as anyone, and I can sympathize with the desire to secure the return of a historical artifact associated with one of the most significant events in Texas history. But this is a matter that should be let alone. The Alamo's sequel was San Jacinto, where Texas won its war, and got independence. Another war followed, in which the US occupied Mexico City, and negotiated at gunpoint the transfer of a huge chunk of Mexican territory for a piddling sum. In 1854, President-General Santa Anna, victor of the Alamo, loser of San Jacinto and Mexico's curse, completed his country's humiliation by agreeing to the Gadsden Purchase, which sold another chunk of Mexico, this one the size of Scotland, to the USA.
However, that was all in the future. On 6 March 1836, the Mexicans won the Battle of the Alamo, capturing the old mission, and the flag of the 1st Company of the New Orleans Greys. The Greys died, but their souls are with God, and may rest easily, because their cause ultimately prevailed. Texas won everything worth winning and the blessed place we live in today is a sufficient monument to their sacrifice. Coveting is a bad business, and we don't need that banner back, not this way. The Mexicans paid for that flag, and we should not insult the memory of the Mexican dead by offering to pay their descendants for a prize won in a fair fight and bought with blood.
It would be pleasing for the flag of the New Orleans Greys to, someday, come home. But it is for the Mexicans, on their own, to come to that decision. Meanwhile, let the dead rest.