It is very difficult to use the records of the General Register Office to trace descendants, rather than forebears, of a particular family. As pointed out above, the birth indexes after 1902 do record the mother's maiden name, as well as the name and surname of the child, so that it can be possible to trace all the births of a particular family from that date forward.
Uncovering the subsequent marriages of those children without knowing the names of their spouses is a much harder propostion, however. To take one example, the likely range of years of marriage for a Michael O'Brien born in 1905 would be 1925 to 1940; there are certainly hundreds of marriages recorded in the indexes under that name.
One could, of course, purchase copies of all of the original register entries in the hope that one entry might show the relevant address and father's name, and then investigate births of that marriage, but in most cases, the work involved makes the task impractical. There are, however, other ways of tracking descendants through land, census, voters' and, sometimes, parish records.