Grashiski appears to be a phonetic spelling, i.e., spelled as it sounds. Lithuanian uses the letter "s^" for the "sh" sound (Polish uses "sz"). My hunch is that it is actually Graz^is^kiai (pronounced grah-zhish-kai) which could easily sound like Gras^is^kiai. Unfortunately, there are several villages with this name in the following districts:
I found a passenger manifest that is very likely that of Mary and her son, William. Very many Lithuanian immigrants to America who used the name, William, were actually named Vincas or Vincentas (for reasons no one I know can explain):
New York Passenger Lists, 1820-1957
Name: Wincas Andzulis
Arrival Date: 2 Aug 1909
Birth Year: abt 1908
Birth Location: Russia
Birth Location Other: pilwisk
Age: 10 Months
Port of Departure: Rotterdam
Port of Arrival: New York, New York
Ship Name: Rotterdam
Wincas (Polish spelling for the Lithuanian Vincas) came with his mother, Maryanna (Marijona in Lithuanian), age 24. Their last residence was "Starapole" which is more than likely Marijampole (which was also known as Kapsukas, Mariampol’, Mariampole, Marjampol, Pas^es^upiai as well as Starapole.) This would be one of the largest towns in the Suwalki gubernia (Russia ruled the lands we know today as Lithuania from 1795 to 1917).
They were born in "pilwisk" which is very likely pilwisk Pilvis^kai in the Vilkavis^kis district, which would have been in the Suwalki province, which is also the province or gubernia noted on the manifest. Graz^is^kiai is about 25km (about 15 miles) due east of the city of Marijampole. It is about 50km (about 31 miles) southheast of Pilvis^kiai.
It was not at all unusual for a woman name Marijona to use either Marie or Mary or even just Ann or Anna when in America.
There was no "usual port" that emigrant Lithuanians used. ALl of them were on the Atlantic coast, the farther south on that coast, the longer the season for traversing the ocean. The most common ports for Lithuanians as well as other eastern European emigrants were Hamburg, Bremen, and several ports in the Netherlands.
Their passage was paid for by the husband and father, "Josas", i.e,. Joseph, living in Forest City, PA, which is where their destination was. So it appears that Joseph arrived prior to his wife and child -- not at all unusual.
There is a good chance that the following record of naturalization is that of Joseph, William's father:
U.S. Naturalization Records - Original Documents, 1795-1972 (World Archives Project)
Name: Joseph Andzulis [Joseph Andzukis]
Birth Date: 23 Mar 1886
Birth Location: Lith, Russia
Children: Lilliam [should be William]; Bertha; Annie; Helen; Eleanova; baldin
Arrival Year: 1906
Issue Date: 29 Oct 1921
Locality, Court: Middle District of Pennsylvania, District Court
He was living at 145 Lackawanna Ave., in Forest City, PA and said he arrived in the U.S. in 1906 from the port of Rotterdam, though he did not know the name of the vessel. The Declaration of Intention names the oldest child as William.