SQL Server CONTAINS ignores single quote before asterix
CONTAINS is used for full-text searching, and can search for:


for a word or phrase
the prefix of a word or phrase
a word near another word
a word inflectionally generated from another word, such as spoon and spooning
synonyms


Perhaps you should use:

DECLARE @String2 NVARCHAR(20) = '"SomeName''%"';
SELECT * 
FROM dbo.NamesTable NT
WHERE NT.Name LIKE @String2;