DNS NS Records vs A Record

There are two distinct options:

1. They can continue hosting DNS for the domain and point the A record to you. This is simplest and they may wish to retain this control. But it introduces some administrative overhead because you will need to contact them if you ever need to change your IP address.

2. They can delegate the entire domain to you. They will need to contact their registrar and ask for the nameservers to be re-delegated to your own. You can't just change the NS records within the zonefile. You will then need to host the domain's zone on your nameservers. This puts you in the most control and you can configure the A record at will.

   However this may also have some administrative overhead, if they maintain other records within the zone (including MXs) and need to contact you whenever they change. You will also need to ensure that you reproduce the entire contents of their current zonefile in order to maintain any services they have today.

Please excuse me for being very blunt, but if you don't know the answer to this question, you should not be running your own name server, let alone hosting NS records for someone else.

In the circumstances the only sane thing for them to do is to only point the A record at your server. They should leave their NS records with someone who understands DNS.

I think there is a third option in addition to the two that Dan suggested, which is more appropriate in some cases when you want to serve someone else's subdomain on your servers, and you want to be able to change the IP address of your server without asking them to update their NS records....

They can delegate just the subdomain that you want to manage to your nameservers. I will quote from dyn.com:

"In your current zone file for your domain at the other DNS provider, you will need to create NS (NameServer) records. These NS records “redirect” queries for the specific subdomain to Dyn Standard DNS. For dynamic.example.com, these records would look like:

dynamic.example.com.    48600   IN  NS  ns1.mydyndns.org.
dynamic.example.com.    48600   IN  NS  ns2.mydyndns.org.
dynamic.example.com.    48600   IN  NS  ns3.mydyndns.org.
dynamic.example.com.    48600   IN  NS  ns4.mydyndns.org.
dynamic.example.com.    48600   IN  NS  ns5.mydyndns.org.

Please note the trailing “.” after each hostname.

Once these NS records have been created, queries for the subdomain will be answered by your Dyn Standard DNS zone.

Good comments here on hosting your own DNS. I think it's a decision you should not take lightly. Due to the hierarchical, cached nature of DNS, a few minutes of misconfiguration or downtime can cause headaches for a long time, as your customers' ISP DNS server might ignore your TTL value and cache a bad response from you for hours. And as DNS issues impact mail delivery to the domains you are authoritative for as well, the peril can be even higher.

See also this related answer of mine to a similar question: Should we host our own nameservers?