After editing the code, enter your DNSimple domain name for Hostname, Account API token for password, Record ID for ZoneID, and TTL.

/etc/inc/services.inc
Add dnsimple to the list of provider values and descriptions:

define('DYNDNS_PROVIDER_VALUES', 'citynetwork cloudflare custom custom-v6 dhs dnsexit dnsimple dnsomatic dyndns dyndns-custom dyndns-static dyns easydns eurodns freedns gratisdns he-net he-net-v6 he-net-tunnelbroker loopia namecheap noip noip-free ods opendns ovh-dynhost route53 selfhost zoneedit');
define('DYNDNS_PROVIDER_DESCRIPTIONS', 'City Network,CloudFlare,Custom,Custom (v6),DHS,DNSexit,DNSimple,DNS-O-Matic,DynDNS (dynamic),DynDNS (custom),DynDNS (static),DyNS,easyDNS,Euro Dns,freeDNS,GratisDNS,HE.net,HE.net (v6),HE.net Tunnelbroker,Loopia,Namecheap,No-IP,No-IP (free),ODS.org,OpenDNS,OVH DynHOST,Route 53,SelfHost,ZoneEdit');

/usr/local/www/services_dyndns_edit.php
In function _onTypeChange(type) add case "dnsimple": just below case "route53" to unhide the needed HTML elements:

...
case "route53":
case "dnsimple":
	document.getElementById("_resulttr").style.display = 'none';
	document.getElementById("_urltr").style.display = 'none';
...

/etc/inc/dyndns.class
In if ($forceUpdate == false && $this->_detectChange() == false) add case "dnsimple": to the same list of all the other providers:

...
case 'citynetwork':
case 'dnsimple':
	$this->_update();
	if($this->_dnsDummyUpdateDone == true) {
...

In function _update() … switch ($this->_dnsService) add:

...
case 'dnsimple':
	/* Uses DNSimple's REST API
	   Requires username and Account API token passed in header
	   Piggybacks on Route 53's ZoneID field for DNSimple record ID
	   Data sent as JSON */
	$needsIP = TRUE;
	$server = 'https://api.dnsimple.com/v1/domains/';
	$token = $this->_dnsUser . ':' . $this->_dnsPass;
	$jsondata = '{"record":{"content":"' . $this->_dnsIP . '","ttl":"' . $this->_dnsTTL . '"}}';
	curl_setopt($ch, CURLOPT_HEADER, 1);
	curl_setopt($ch, CURLOPT_CUSTOMREQUEST, "PUT");
	curl_setopt($ch, CURLOPT_HTTPHEADER, array('Accept: application/json','Content-Type: application/json','X-DNSimple-Token: ' . $token));
	curl_setopt($ch, CURLOPT_URL, $server . $this->_dnsHost . '/records/' . $this->_dnsZoneID);
	curl_setopt($ch, CURLOPT_POSTFIELDS, $jsondata);
	break;
...

In function _checkStatus($ch, $data) … switch ($this->_dnsService) add:

...
case 'dnsimple':
	/* Responds with HTTP 200 on success.
	   Responds with HTTP 4xx on error.
	   Returns JSON data as body */
	$header_size = curl_getinfo($ch, CURLINFO_HEADER_SIZE);
	$header = substr($data, 0, $header_size);
	$body = substr($data, $header_size);
	if (preg_match("/Status: 200\s/i", $header)) {
		$status = "phpDynDNS ({$this->_dnsHost}): (Success) IP Address Updated Successfully!";
		$successful_update = true;
	} else if (preg_match("/Status: 4\d\d\s/i", $header)) {
		$arrbody = json_decode($body, true);
		$message = $arrbody['message'] . ".";
		if (isset($arrbody['errors']['content'])) {
			foreach($arrbody['errors']['content'] as $key => $content) {
				$message .= " " . $content . ".";
			}
		}
		$status = "phpDynDNS ({$this->_dnsHost}): (Error) " . $message;
	} else {
		$status = "phpDynDNS ({$this->_dnsHost}): (Unknown Response)";
		log_error("phpDynDNS ({$this->_dnsHost}): PAYLOAD: {$body}");
		$this->_debug($body);
	}
	break;
...