Perhaps this is your grandmother - Rebekka Tobiassen on Hommen in Lyngdal, Vest-Agder, Norway in 1910 census with a very large family, father Tobias:
http://digitalarkivet.arkivverket.no/ft/person/pf01036598001...The census is distinctive in including family members living away - there are 5:
Andreas Tobiassen is in Minnesota
Elisabet and Rebekka are in Brooklyn, NY
Martin is also in New York
Torvald is in Minnesota
This Rebekka was born 09.09.1888.
Parents are Tobias Abrahamsen Hommen and Gunhild Olsdatter.
Probably this is Rebekka's father Tobias Abrahamsen, Hommen in 1865:
http://digitalarkivet.arkivverket.no/en-gb/ft/person/pf01038...A bygdebok is a local history book. On this thread, a poster has checked the bygdebok for this area, where it is written that Inger Rebekka Tobiasen Hommen married Olav Lien:
http://archiver.rootsweb.ancestry.com/th/read/NORWAY/2001-02... A little digging might uncover the reason for Haeme. Maybe Hæme was an old spelling of the same farm name? Or it's a separate farm. Siblings could and did have different farm names (address names) from each other.
The Original Poster supplies too little info for us to know who Thomas Haeme is - no locations, no dates, full original name unstated. More than one unrelated families could live on same farm and use that farm name in the US as a surname. The Original Poster seeks info on "my grandfather" - I have a family member, a young mother, whose gradfather was born in 1950 and a family member, a retired woman, with a grandfather born in 1850. So 'grandparent" can encompass a wide range of birth years. But yes, it could be the same family.
Here is an arrival at Ellis Island of Thorvald Tobiassen, Oline Tobiassen, and Martin Hommen:
http://174.143.111.10/shipping/FormatTripPass.asp?sship=Kris...=
If you can find Rebeka's arrival, the passenger manifest will tell where she's from. Ditto her marriage record, death record, obit, or interviews with your family elders.
Here is a Find a Grave memorial for Rebekka Lian (1888-1970) buried Oak Hill, Greene county, New York. Since your user name includes "lian" I suspect this is your grandmother and she was indeed born 1888.
http://www.findagrave.com/cgi-bin/fg.cgi?page=gr&GRid=13... Hope this helps.
-- a volunteer researcher
not related