There is a general rule that for a kanji compound of the form 大＊, you tend to read 大 using 音読み (タイ・ダイ) if ＊ is 漢語 and with 訓読み (おお) if ＊ is 和語. See e.g. this question on goo (in Japanese). (Of course there are many exceptions, and this rule is not special to 大---see General guidelines for choosing 訓読み vs. 音読み in kanji reading)
Are there any rules of thumb or broad etymological reasons for which 音読み to use for 大 at the beginning of a kanji compound?
For instance, you typically read the following examples as タイ＊:
and in these as ダイ＊
One subjective impression I have is that if you have some 漢語, at least in the case of 二字熟語, and modify it with 大 at the beginning that it is typically read as ダイ, e.g., 大人気 or 大家族. (Compare this with 大使館, say, where the root word seems to be 大使.)