The most sophisticated alternative theory is perhaps that of Florentino Ameghino who postulated an autochthonous evolution of species based on skeletal anthropology and diggings according to the following chart:
German anthropologists, led by Hermann Burmeister, opposed this theory. Burmeister proposed a catastrophe theory of human evolution, similar to that of Cuvier. According to these ideas people evolved not gradually, but very rapidly and independently. William Henry Holmes was active in disputing that there was any evidence humans were in the Americas before the last glaciation, and therefore must have come from elsewhere. Franz Boas disputed this and argued that man came in the interglacial period. Likewise, Daniel Garrison Brinton argued that American man must have arrived in the glacial period and since Siberia was thought to be uninhabited at that time, they could not have a Siberian origin, but must have been Polynesian or some other Asiatic emigration. Brinton also argued that Eskimoes are completely distinct from Siberians, yet occupy the entire Arctic circle, so they and other Americans, who are akin to them, must have had an independent, non-Siberian origin. Also, he disputed that the idea that America was cutoff because Eskimoes circumnavigate with no need for a land bridge.
Prior to these theories, there was a large school of thought that considered Americans to be autochthonous. These theorists included: Levegue, Humboldt, McCullogh, Morton and Quatrefages. Most of these ideas proposed only semi-autochthony and had some catastrophic or evolutionary component. A few theorists, such as Kaimes, Morton, Nott and Glidden supported fully autochthonous origins with no old world component. Note that many old theories included over-land travel, so the idea that a "land bridge" is a modern idea is somewhat incorrect. The theme that native Americans derived from Siberians of some kind or Mongolians was prevalent very early on.
Just as a side note, the "theories" that Indians derived from Israelites or Phoenicians or some such, were just fantasies by "historians" writing for a public audience, and were not considered serious science by real anthropologists. I have listed above the main scientific thinkers from the pre-modern period who were considered serious scientists.