Does the fact she is living in an immigration flat (I've never heard of such a thing) mean she is awaiting deportation? If not, how come she is there, as it seems from what you say she has no present legal basis for her being in UK?
If she is marrying to get a passport they should be aware getting one is not an easy quick or straightforward process. In most cases the spouse has to apply from outside the UK for permission to enter as a spouse. This - if granted - is for a limited period & permanent residency can only be obtained several years (I think it is at least 2 yrs) later - & depends on the state of the marriage at that time. Also, her child would be a problem unless your son does adopt him/her, or unless the child has British citizenship (probably depends who the father is).
Go here:
http://www.ind.homeoffice.gov.uk/
You will have to search around but should be able to find info. on the spouse situation & maybe the child.
In my view, you should not provide a dowry. If you wish, you could probably find out from the Embassy of her home country (address & contact details should be on the net) whether provision of dowries by the bridegroom's parents is normal practice in her country.