Database does not show whan adding ODBC connection.

Did you restore the database from a different server? I am thinking the logins may be out of sync. Here is a script that I use to test for logins that are out of sync. The first query looks for the logins and the second query syncs all the logins.

--Query 1 to test for logins.
DECLARE @Collation varchar(100)
DECLARE @SQL VARCHAR(2000)

CREATE TABLE ##TempSync
(
DB_NME Varchar(50),
DBUserName varchar(50),
SysLoginName varchar(50)
)

SELECT @Collation = CONVERT(SYSNAME,DatabasePropertyEx('master','Collation'))

SET @SQL = 'USE [?]
SELECT ''?'' DB_NME,
       A.name DBUserName,
       B.loginname SysLoginName
FROM sysusers A
      JOIN master.dbo.syslogins B
      ON A.name Collate ' + @Collation + ' = B.Name 
      JOIN master.dbo.sysdatabases C
      ON C.Name = ''?''
WHERE issqluser = 1
       AND (A.sid IS NOT NULL
       AND A.sid <> 0x0)
       AND suser_sname(A.sid) IS NULL
       AND (C.status & 32) =0 --loading
       AND (C.status & 64) =0 --pre recovery
       AND (C.status & 128) =0 --recovering
       AND (C.status & 256) =0 --not recovered
       AND (C.status & 512) =0 --offline
       AND (C.status & 1024) =0 --read only
ORDER BY A.name'

INSERT into ##TempSync
EXEC sp_msforeachdb @SQL

SELECT * FROM ##TempSync

DROP TABLE ##TempSync

--Query 2 to Sync Logins

DECLARE @Collation VARCHAR (100)
DECLARE @SQL VARCHAR(2000)

SELECT @Collation = CONVERT(SYSNAME,DatabasePropertyEx('master','Collation'))

SET @SQL = 'USE [?]
DECLARE @DBUserName varchar(50)
DECLARE @SysLoginName varchar(50)
DECLARE SyncDBLogins CURSOR FOR
SELECT A.name DBUserName,
        B.loginname SysLoginName
FROM sysusers A
      JOIN master.dbo.syslogins B
      ON A.name Collate ' + @Collation + ' = B.Name 
      JOIN master.dbo.sysdatabases C
      ON C.Name = ''?''
WHERE issqluser = 1
       AND (A.sid IS NOT NULL
       AND A.sid <> 0x0)
       AND suser_sname(A.sid) IS NULL
       AND (C.status & 32) =0 --Loading
       AND (C.status & 64) =0 --pre recovery
       AND (C.status & 128) =0 --recovering
       AND (C.status & 256) =0 --not recovered
       AND (C.status & 512) =0 --offline
       AND (C.status & 1024) =0 --read only
ORDER BY A.name

OPEN SyncDBLogins
FETCH NEXT FROM SyncDBLogins
INTO @DBUserName, @SysLoginName

WHILE @@FETCH_STATUS = 0
BEGIN
    EXEC sp_change_users_login ''update_one'', @DBUserName, @SysLoginName

    FETCH NEXT FROM SyncDBLogins
    INTO @DBUserName, @SysLoginName
END
CLOSE SyncDBLogins
DEALLOCATE SyncDBLogins
'
EXEC sp_msforeachdb @SQL