Are you looking for team names appropriate for 1871, or modern-style names that
sound appropriate for 1871? Since most teams in 1871 didn't have formal names, an appropriate name for a baseball club would be "the base ball club." For instance, the team from Denver would be known formally as "the Denver base ball club." Informally, the team would probably be called "the Denvers." If the city itself had a nickname, the team might adopt that (e.g. the Cleveland Forest Citys, the New Haven Elm Citys). In the case of Milwaukee, it hosted an amateur team around that time called the "Cream Citys."
If the team members wore a distinctive article of clothing, the team might be nicknamed for that (e.g. the Cincinnati Red Stockings, the Hartford Dark Blues, the Worcester Ruby Legs, the Baltimore Canaries). That's an example of a
synecdoche. Otherwise, if the team was formed from members of a sports club, it might adopt the club's name (that's how the Philadelphia Athletics got their nickname - they originally represented the Philadelphia Athletic Club).
It was only later - much later - that teams formally adopted nicknames. Those nicknames often originated in the newspapers as an alternate way to describe the club. For instance, the Cleveland team had a bunch of skinny, long-legged players, so the newspapers began referring to them as the "Spiders." Brooklyn had a sudden spate of marriages among its members, so reporters started calling them the "Bridegrooms." Those names, however, were strictly unofficial, and the newspapers would switch to a more imaginative moniker when the occasion arose. For instance, the Chicago team was known for many years as the White Stockings. Then, when the team fielded a bunch of young players, the newspapers started calling them the Colts. Then long-time player-manager Cap Anson left the club, and it was known as the Orphans. Then the team got another bunch of youngsters, and the papers started calling them the Cubs. That last one stuck.