the Familysearch Civil index that aghadowey mentioned is the best way to track Civil registrations. I had a quick look and based on the census details found a few matches to get you started.
1. Probable marriage of Henry :
Name: Henry Bovenizer
registration district: Dublin South
quarter / year: Oct - Dec 1890
volume: 2 / page: 523
The Dublin South registration district includes both the southern part of the city and suburbs including Rathmines. There's a possible cross-match for this marriage entry with an Annie James (same quarter, district, volume & page)
2. possible birth for Henry
Name: Henry Bovenizer
Reg. district: Rathkeale
year: 1871
vol: 5 / Page: 726
If this is correct then his age is out by a little bit on the census.. but the district of Rathkeale matches the county of birth (Limerick) given on the census
3. birth of Elizabeth Mary :
name: Elizabeth Mary Bovenizer
registration district: Dublin South
quarter / year: Apr - Jun 1891
volume: 2 / page: 705
4. birth for Annie (jnr)
Name: Annie Bovenizer
reg. district: Dublin South
quarter & year : Oct - Dec 1901
volume: 2 / Page: 547
You can order certs from the GRO using the above details - see
www.groireland.ieShane