Since most immigrants came to America via ship and space was very limited, many brought only the types of items that would help them survive in the New World. Knowing that they may never see their homeland again, some brought the personal items they valued most.

Families typically left large items such as furniture behind simply because it wasn't practical or affordable to try and transport them across sees. Important pieces of clothing and valuable items that were also useful, such as china (dishes) and linens (tablecloths) or pots and pans, were common brought. Small trinkets and jewelry were also brought along, not only because they were important to the immigrants but also because these items could be sold if the family found itself in need of money.

Culture also frequently played a role in the choices immigrants made in deciding what to bring with them. People might leave some items behind in favor of those that were important to the practice of their religions.

~ From Reference.com

From Smithsonian Magazine:

"... For most immigrants who didn't travel first- or second-class, the sea voyage to the United States was far from a cruise ship with lavish buffets. Passengers in steerage survived on "lukewarm soups, black bread, boiled potatoes, herring or stringy beef." Josephine Orlando Saiia, of Greenfield, Wisconsin, contributed a recipe for an Italian cookie called mustasole that she says sustained many immigrants on the long voyage, because they "are very, very hard when dry and become chewy when damp—like an ocean voyage. They do not spoil, can be eaten for a year, keep well with no crumbs... I have one that's fifty years old."

From Quora

Immigrant passengers, unlike prisoners, could and did bring some of their own foods. Exactly what these were is unknown, but would have included local fruit, sausages, cheeses, and breadstuffs, at least until they ran out. They would vary, too, according to where the immigrant was traveling from. Eastern European passengers, for example, would likely have brought some sort of sauerkraut or other pickled vegetables. On trans-Atlantic travel, salt beef, salt pork, and salt fish would have been likely additions as well. These formed a major part of merchant marine diets and would likely have been part of passenger diets, too.