A couple, whose wedding I officiated at a number of years ago, had a son two years ago. He was circumcised by a mohel but at the pediatrician’s office as a procedure. No formal ceremony happened. Now the parents want to give their son a Hebrew name. While naming usually happens at the bris, is there any way to give this boy a name without another ceremony (hopefully not hatafat dam brit or anything similar)?. Is there anything to do b’di’eved? The parents very much want to connect their child more formally to the Jewish community and both parents are Jewish.
Giving a name does not require a bris or a ceremony (think of when we name a girl). All that is required is that the child is called by that name. I would say that the father should get an aliyah and have the gabbai make a mishebeirach which will mention his wife and his son (by his new name). Also, if they are actually going to call him that new name in practice, just starting to do so in the presence of others will suffice.
As to the bris—not that you asked, but if it was done prior to day eight, then it was not a kosher bris. The child is still not an arel, but if this is something that they care about, then a hatafat dam brit would be what would be recommended (Arukh HaShulkhan YD 262:5). Although, I should point out, that it is debatable if this actually allows one to fulfill the mitzvah once the foreskin was removed. It might simply be “מעוות לא יוכל לתקון”