Q:

Was there a serious proposal to offer India territory in East Africa?

In the recent World Service series The War That Changed The World the episode dealing with India contains a rather extraordinary claim. 
A panel member, an Indian, says there was a notion during WW1 that India might get her "own" colony in German East Africa as a prize for her substantial contribution of troops and money to the British war effort.
Perhaps not so extraordinary, Several British Empire dominions did receive their own colonies or mandates. Australia, South Africa and New Zealand helped carve up Germany's colonial possessions.
The notion that India could gain her own overseas empire ... were there any serious proposals?

A:

Yes, surprising as it is, I found credible sources indicating that there was some discussion of offering India East Africa as a mandate.
Perhaps it is useful for others who wish to read more about this a full detail of my sources. In "How India Became Territorial: Foreign Policy, Diaspora, Geopolitics (2014)" by  Itty Abraham, I found this quote:

On war reparations, in spite of their considerable contribution to the
  war effort, the opinion of lesser powers and British dominions would
  count for little. Powerful elite Indian opinion had proposed that India
  be permitted to take over the German colonies in East Africa as a
  mandate, but that did not happen.

Abraham references "India and East Africa: Imperial Partnership at the End of the First World War (1971)" by Herbert Lüthy, adds some important context:

It is in this context that the even more curious question of a League
  mandate for India came up, rather casually, for deliberation in the
  British War Cabinet in 1918, and again in the Mandates Commission on
  African Colonies at the Paris Peace Conference. With the exception of
  Canada, each British Dominion was ultimately entrusted with mandatory
  powers over some part of the colonial spoils from Germany's lost
  empire; to entrust India, or rather the Government of India, with a
  mandate over former German East Africa might have added some more
  substantial bribe to the purely symbolic promotion to League
  membership, although - quite apart from all the incalculable
  consequences we can envisage from hindsight - it would equally have
  added a further paradox to the ambiguity of imperial and international
  status. League membership for India had been difficult enough to
  admit, but had been allowed for as making simply 'an additional voice
  for the British Empire' (the founding of the United Nations witnessed
  a similar allowance for another great power), but a colonial mandate
  for India would have been more than even the most subtle casuists of
  international law could stomach.

He goes on to quote "The International Status of India (1931)" by Lanka Sundaram, which provides the closest secondary evidence I could find of this discussion:

The League Mandate for the administration of German East Africa (now
  Tanganyika) was at one moment on the point of being granted to India,
  but this courageous step, which would have enhanced the value of the
  juridical basis of India's international status, was at the last
  moment withdrawn in favour of Great Britain.

Finally, I did manage to dig up a primary source document indicating the same thing. In the "Minutes of the 37th Imperial War Cabinet, Minute 8" (a 50 MB PDF), the British clearly intended to at least allow India to make its wishes known with respect to German East Africa.

Sir Robert Borden said that there were still a further question on
  which it was possible the British Government would differ from the
  American Government, and this question was with regard to the
  retention of the German colonies. Sir Robert reminded the Imperial War
  Cabinet that  he had already suggested by telegram that those
  dominions who were principally interested in this question should put
  forward their views... 
... the view of the British Government was that none of the German
  colonies should be restored, and that those German colonies which had
  been captured by colonial troops, with the possible exception of
  British East Africa, should be held by the dominions which had
  captured them. 
... the Imperial War Cabinet decided that - At the important Allied
  Conference which should precede the Peace Conference, India and each
  self-governing dominion should be given the fullest opportunity to
  express their views on those questions which may closely concern them.

I think that the evidence strongly indicates that such an offer was at least (perhaps disingenuously) discussed.

