I sympathise with you umalig.
I believe that your son not being recognised as an Irish citizen is unlawful, though I am not an expert. I see nowhere in the legislation posted in the link above, nor anywhere else on the Irish Naturalisation and Immigration Service website, that periods spent in Ireland as a student on a student visa are not considered as lawful residence. Perhaps someone else can confirm.
Regarding your son getting a Lithuanian passport. I'm assuming that he is a Lithuanian citizen by decent, passed on by his mother as she is a Lithuanian citizen at the time of your son's birth. I am not sure of this because I am in no way familiar with Lithuanian legislation on this matter, however I'm pretty sure that this should be the case. In order to get a Lithuanian passport, does your son and your wife really have to go to Lithuania? Surely this can be obtained from the Lithuanian embassy in Dublin?
I must stress that regardless of your son's nationality, it is very likely that residence for yourself will be refused by the DoJ because you have not previously lived in another EU country with your wife / son. I know that you are concerned that, as a direct result the the DoJ's unlawful interpretation of EU Directive 2004/38/EC, you are being put in a position that makes it unlawful for you to work and thus provide for your young family. This is why I am suggesting to you the idea of applying for a UK EEA Family Permit at the British embassy in Dublin and moving (at least for 6 months) to Northern Ireland. Understandably, you may find the idea of uplifting the three of you and moving (effectively) to another country may be daunting, but it's really not too far away and you will at least be legally resident and able to earn an honest crust to support your wife and child. All the best to you, umalig.
