--What is that what-you-might-call-barrel on that platform by the side of the house?

For a sentence like this 'dersin' is correct which means what do you call .... ?
or you can also say 'diyebilirsin' which means 'what you might call'

My wedding present to him is going to be what you might call...... his trousseau Ona büyük bir düğün hediyesi vereceğim.Çeyiz de denebilir.

For this one, 'denebilir' means 'it might be called'

Some of us are what you might call svelte Bazılarımız böyle narin yapılıdır.

Bazılarımız, narin diyebileceğin yapıdadır.
This translation would be better because in this sentence the main verb of the sentence is 'yapıdadır' and the suffix -acak is a gerund suffix over there.
And the same explanation can be used for the last sentence.

--------------------------------------------------
Note added at 1 hr (2012-07-11 10:49:04 GMT)
--------------------------------------------------

If the sentence is without "what is that" than the answer is
Evin yanındaki platformdaki varile ne diyebilirsin?

If it is with what is that, than the answer of Levent Oğuz is correct.

Evin yanındaki platformda varilde diyebileceğin şey ne(dir) ?

--------------------------------------------------
Note added at 23 hrs (2012-07-12 08:31:02 GMT)
--------------------------------------------------

by the way 'de' is seperate : 'varil de '