Indicates the barrels were made by twisting wire around a mandrel then welding.
Yes, that is a very simple explanation of how Twist steel barrels were manufactured; but does not answer the original posters question of "What does 'New Twist' mean on the top of a 12 gauge Syracuse Arms shotgun?"
Prior to, and for several years after the turn of the last century; most shotgun gun barrels were made by wrapping alternating strips of iron and steel (called ribands) around a steel bar called a mandrel. These alternating bands of iron and steel were heated to a red-hot state; then, as workers wrapped these ribands around the mandrel, other workers pounded these two materials with hammers so the the combination of heat and force fused them together to form a hammer weld.
The simplest, and least expensive barrel tubes made in this manner were called "Plain Twist"; which tubes consisted of one untwisted riband of alternating steel and iron. Barrel prices went up from that point, as the makers of these barrel tubes were very skilled at twisting these ribands into elaborate patterns; which "patterned" ,or figured barrels were then named "Damascus". The lease expensive Damascus barrel tubes might have a pattern created by twisting together only two ribands; with the cost increasing in direct relationship to the numbers of ribands used, and the number of times these ribands were twisted together. These barrels were often referred to as "figured steel", with the figure, or pattern being revealed by the "browning" or barrel finish process; the iron bands would finish darker, color wise, that than the steel bands, allowing the intricate whorls and curls to be revealed. The makers of these barrels created some incredibly intricate and fine Damascus steel barrels, with the master makers so skilled that they learned to twist these iron and steel ribands into words. There is one example of a Remington Grade EE 16-bore double that has the word "Remington" repeated over and over throughout the full length of each barrel tube; and there is a Belgium maker, Pieper, who ordered a set of Damascus barrels featuring the Pieper name for the full length of those special barrel tubes.
With guns manufactured by the Syracuse Arms Company, the type of barrel steel used indicated gun quality and price. The term "New Twist" was the barrel steel name the company assigned to the Plain Twist barrel tubes used on their lowest grade double barrel shotguns; which name began to be roll-stamped atop the barrels around serial number 8,000 (earlier models have no barrel steel identifiers). Most Syracuse guns having New Twist barrels will be Grade O guns ($30 suggested retail, with the most expensive Grade D SAC gun retailing at a whopping $475.00!). SAC grade stamps will be found on the water table of the frame, and also on the barrel flats; but New Twist barrels can/will also be found on the Syracuse Grade 1 gun (discontinued around 1896, and identical to the Grade O gun in materials and finish, but with a small amount of scroll engraving on the gun frame; the Grade 1 retailed at $35), and on a few of the seldom seem Syracuse hammer double guns made in late 1904 and into early 1905 when SAC gun production ceased.
Unless one of the rare high grades, or a low-grade that remains in virtually new condition; Syracuse guns have little collector interest or value, and the Grade O gun is by far the most common with surviving examples seldom bringing more that $200-250 at auction. That said I have seen one Grade O SAC gun that remained 95% new, and the factory finishes on that gun were stunningly beautiful; based on that example it was no mystery as to why the company sold so many Grade O guns.