lui et leur

Question

In the second sentence, the conversation has "lui" although it is hard to distinguish "lui" from "leur" with the speaker's intonation.  For the remainder of the conversation, the conversation has "leur" when referring to the recipient(s) of the gift.  While I can get the difference after listening for multiple times, I still find it strange that the two are not consistent.

Accepted Answer

There is a mix of:
1. 'Leur' as the possessive adjective - leur petit Arthur, leur nouvelle vie - translating as "their"
and 
2. 'Leur' as the indirect object pronoun - 'faire plaisir (à)' and 'offrir (à) - translating as "them". 
'Leur' refers to the parents (although in the last case, it can also include Arthur as it is ' leur nouvelle vie à trois '. 
3. 'Lui' as the indirect object pronoun - ' trouver qqc pour qqn ' - translating to him. 
The conversation is acknowledging that you might buy a present 'for' a newborn, but it is really to ' give (make) pleasure for the parents ' , and it is ' given (offered) ' to the parents - even more so here, as the proposed present will be of no interest to Arthur in the near future !
In terms of hearing the difference, it will come over time - both recognition of the difference in pronunciation (for instance, the French 'r' comes through on the end of leur, and is definitely not present with lui) and context recognition improve over time.