Dear Mr. Delaney,
Thank you for posting your request on History Hub!
We searched the Access to Archival Databases (AAD) and located the Records for Passengers Who Arrived at the Port of New York During the Irish Famine, documenting the period 1/12/1846 - 12/31/1851 in the Records of the Center for Immigration Research (Collection CIR) that includes 279 partial records for individuals with the last name Delaney arriving after January of 1850 that may include your ancestors.
We searched the Bureau of Land Management (BLM) database that provides 3 hits for the last name Delany between 1850-1865 in the Minnesota territory. If these are ancestors of yours, this would be an indication of why they traveled to this unsettled territory.
We also suggest reviewing the National Archives blog Pieces of History post: Exploring Irish History Through the Famine Records and Other Arrival Records for useful information about the motivation of over 600,000 individuals leaving Ireland during the period of 1846-1850.
Lastly, we searched the database Chronicling America and located 6520 articles pertaining to land in Minnesota in various newspapers published between 1850 and 1868 that may provide information about why they moved to the Minnesota Territory.
We hope this is helpful. Best of luck with your family research!