How are people finding the name of the ship their ancestor came to New York on?

Basically you have to look at passenger arrival lists. If they arrived 1892 or later, look for Ellis Island records. Prior to that the port of NYC was Castle Garden.
You did not give any specific information about Adam Bohn (birth year, arrival date, place of birth, etc) so it is difficult do determine which Adam Bohn is the person you are looking for.
I found 3 arrivals for Adam Bohn in the late 1800's:
Adam Bohn, 38, farmer, from Dermbach, Prussia, headed to NY, arrived NY on SS Dresden 23 Aug 1892
Adam Bohn, 5 or 6, from Bavaria, traveling with mother and several other children, arrived NY on SS Zeeland on 21 Oct 1882
Adam Bohn, 17, shoemaker, arrived NY on SS Schiedam 26 May 1884
There were 2 other Adam Bohns, who each were from Russia and were immigrating with their wife and children.

You could search at https://stevemorse.org/index.html the Gold Form for 1892 to 1924 and the White Form for 1820 to 1957.

You might also look for naturalization documents. Applicants were asked the ship on which they traveled.

 LittleDocsdaughter 

Thank you for posting your question on History Hub!

Passenger arrival records at the National Archives and Records Administration (NARA) document the arrival of citizens and immigrants into U.S. ports between approximately 1820 and 1957. The U.S. Customs Service maintained passenger lists from 1820 through 1890, although these records are not always complete. Beginning in 1891, the Immigration and Naturalization Service (INS) maintained records of arrivals.

Until January 1, 1820, the U.S. Federal Government did not require captains or masters of vessels to present a passenger list to U.S. officials. Thus, NARA generally does not house passenger lists that predate 1820. There are two exceptions: NARA has passenger lists for Philadelphia arrivals beginning in 1800, and for New Orleans arrivals beginning in 1813.

For pre-1820 passenger arrival records for other ports of entry, check with archival repositories in the city or state where the port is located. For additional information about passenger arrival records that predate NARA holdings, see: P. William Filby and Mary K. Meyer, eds., Passenger and Immigration Lists Index: A Guide to Published Arrival Records of ... Passengers Who Came to the United States and Canada in the Seventeenth, Eighteenth, and Nineteenth Centuries, 3 volumes plus annual supplements (Detroit: Gale Research Co., 1981).

NARA’s passenger arrival records are available as microfilm, and the microfilm has been digitized in full and made available online through multiple partner websites. For information about these resources, please consult our webpage https://www.archives.gov/digitization/digitized-by-partners. Please note that researchers can access these online resources free of charge from any NARA facility.

We hope this assists you with your research!

Sincerely,
Archives 1 Reference Branch (RR1R)
RR1R-23-65555-RH

Oh thank you! It is bringing tears to me just thinking I may find information. I will get exact dates re: his b/d, birth place. And take the advice you so quickly gave me

Thank you. I will give that a try!

15, January to 1847 Immenhaussen Darmstatd Germany ; Adam Bohn. I have seen it listed as "Bavaria" Germany