I...I don't think this is an appropriate question; for either an academic sub or any sub for that matter. I take great pleasure in tackling everyday negative myths about WWI and am, at times, even accused of being a war hawk; in fact quite often. However even this is too far for me; ultimately this was one of the most brutal wars in history almost regardless of where you were; even the African front was not saved from suffering with both sides (but especially the Germans) exploiting the Africans within. Just in Paul von Lettow-Vorbecks shenanigans in East Africa alone throughout the war upwards of 400,000 forced African laborers alone would die under his watch.

If you want to be safe your best bet is something far away from anything that could give you "a war story or two". Frankly this is war and there is no safe zone for people to have the time of their lives on an adventure while also being pristine and safe. You're either a paper pusher or some mechanic on a plane away from the action or you are in danger; it's that simple. You're either on the front lines or serving in some capacity on them, on the ships dealing especially with persistent U-boat threats or in a plane about to be shot out of the sky from small arms fire.

If you were pressing me instead where the safest place for a 'frontline soldier' (or people serving on their behalf in other ways) I'd instantly say the Middle East; particularly the Egyptian Expeditionary Force. Lots of maneuver and small battles; I don't know off the top of my head any major battles where the Brits lost anything more than a few thousand men and most major battles consisted of a few dozen to a few hundred out of numerous available divisions. The Western Force as it was appropriately called dealt with Libyan jihadists and German support while the Eastern Force (again appropriately called) held the Suez Canal and then pushed into Palestine.

However with that said this would have been a total anomaly if you were British. The Light Horse brigades were Australian and the Imperial Service Cavalry was from various Indian ethnic groups. The British infantry divisions which were sent were sent from specific localities as British infantry generally was raised in the early days of the war; you couldn't say "I wanna go to the Egyptian party" it was "You and your buddies signed up in Lancashire; we're sending a division of you blokes there (3 brigades) and a division of Scots too."* Generally that's how it worked in these things; most forces in the Mideast and fighting in not-Europe for the Brits were colonials and those who supported them as British mainlanders were just sent there as a cog in the machine.

Wikipedia is generally faux-pas around here but the page on the Egyptian Expeditionary Force gives a great overview of the order of battle of the various forces there throughout the war. As you can see the lions share of heavy lifting was done by ANZAC (Australian/New Zealander) troops with certain British mainlanders cycled in and out. Essentially if you weren't part of those groups raised at the time those groups were raised in those specific regions you could not have "requested" to be there even as an officer. However, with that said, throughout '15 and early '16 a lot of Kitcheners "New Army", Britain's first modern volunteer army, would be cycled into this Mediterranean/Mesopotamian theater to gain battle experience before being sent into the Western Front. So, again, if you were in this 'safe' area you could likely count on being sent to the place you least wanted to go.

If you were French you're shit out of luck; you're on the front line or you're an artillerymen being hit by counterbatteries all day with no fantastic 'war stories'. If you're German more or less the same story; you couldn't really request where to go it was where you were stationed. You could count on either being killed by the Russians or French or British though so it's a tough decision. Let's not even get started on how out of luck you are if you were a Russian conscript though.

TL;DR:

No.

Ok, in terms of being British.

Firstly, if it's 1915, you still have options militarily. Full blown conscription didn't come into being until 27th Jan 1916. In 1915 your best option to get an easy ride was to join the territorials and get shipped off to replace a regular army unit that was being recalled somewhere far flung in the dominions out of harms way.

If you are looking for relative safety above adventure, the trick was to avoid the military and get a career that was 'essential to the national war effort'. Even then, there are towns up and down the coasts of England with longer lists of fishermen and merchant navy killed during the two wars than fighting.

But in reality it depends on what you mean by connections. It's highly doubtful anyone with proper 'connections' would have ended up in the Great War on the British side without being an officer, given the social gulf between the working man and the gentleman on the street was wider than that between the lower ranked officers and conscripts.

Edit to add the Military Service Act of 1916