You suggested an alternative:
Somebody's special kitchen.
This would mean that somebody has a special kitchen, which is not what Tom Hanks had meant to convey.
To explain the phrasing that sounds odd to you, look at it this way:
Somebody special's kitchen.
The possessive applies to the entire noun phrase, not just to the adjective.
Part of the reason it sounds strange, however, is that the possessive apostrophe comes immediately after the adjective. We are used to possessive apostrophes coming immediately after nouns.
Most likely, this would sound better to you:
A special somebody's kitchen.
It's still a noun phrase, but the adjective now comes before the noun.
It could also be rephrased entirely:
In the kitchen of somebody special.
In the kitchen belonging to somebody special.
But, grammatically, the original version is correct—even if it might sound odd.