Dynamic IP-address allocation (central management)
If your Banana-Pi gets its IP-address from a DHCP service (e.g. on your router) then it has the opportunity to inform the DHCP server of its hostname. The DHCP service often has the facility to inform the DNS service (e.g. also on the router) of this. Otherwise your Banana-Pi may be assigned some default hostname.
Static IP-address allocation
If your Banana-Pi has a static IP-address locally configured, it won't ask the DNS server (via DHCP) to add its name to its data.
Fixed IP using DHCP reservation
If you want your Banana-Pi to have a fixed IP-address, you can configure a DHCP reservation on the router and reconfigure the Banana-pi to use DHCP. how you configure a DHCP reservation depends on the router make and model. Administratively, this is the simplest solution.
The router will then always allocate the same IP-address to the Banana-Pi and will update the DNS service with the Banana-Pi's current hostname.
References
3.3. Client Desires Server to Do DNS Updates
A client can choose to delegate the responsibility for updating the
FQDN-to-IP-address mapping for the FQDN and address(es) used by the
client to the server. In order to inform the server of this choice,
the client SHOULD include the Client FQDN option in its DHCPREQUEST
message and MAY include the Client FQDN option in its DHCPDISCOVER.
The "S" bit in the Flags field in the option MUST be 1, and the "O"
and "N" bits MUST be 0.
the Armbian Debian jessie dhclient does not handle a served hostname reliable
option host-name string;
This option specifies the name of the client. The name may or may not be qualified with the local domain name (it is preferable to use the domain-name option to specify the domain name). See RFC 1035 for character set restrictions. This option is only honored by dhclient-script(8) if the hostname for the client machine is not set.
The DHCP server issues a private IP address dependent upon each client's MAC address, based on a predefined mapping by the administrator. This feature is variously called static DHCP assignment by DD-WRT, fixed-address by the dhcpd documentation, address reservation by Netgear, DHCP reservation or static DHCP by Cisco and Linksys, and IP address reservation or MAC/IP address binding by various other router manufacturers. If no match for the client's MAC address is found, the server may or may not optionally fall back to either Dynamic or Automatic allocation.