
Whilst working out coordinates may not be suitable for children, they will certainly enjoy following the clues and finding the cache at the end - provided they don't mind walking!
Before beginning this quest it would be beneficial for you to have a notebook and pen, also Google could be helpful.
Congratulations once again go to "Gibby McGinge" on achieving the First Time Find.

Artists impression of Gibby McGinge attempting to negotiate through the undergrowth with his bike.
Now let's begin......
Clue 1
The mysterious owner of the Black Tower, located somewhere in the New Forest, has hidden an artefact (the geocache) somewhere in Southampton Common (for which he / she apparently obtained permission from Southampton City Council). Follow the clues to discover where it has been hidden. At the coordinates you have been given you will find a large wooden pole, with the Black Tower initials carved into it. Under these initials you will see these numbers:

Fill in the missing numbers from these coordinates: N 50° 56.(X+1)(Y)' W 1° 24.69(Z)' and off you go!
As you walk down the road you may spot the symbol below, which has appeared at numerous places in this road. Its significance is not known, but it certainly seems to indicate Black Tower activity in the area.

Clue 2
The clue is the place you are at: morning or evening? For morning X = 9, for evening X = 5.
Chose wisely and then head for: N 50° 56.08X' W 1° 24.6X7'
Clue 3
Look for a prominent sign, if you can't see one you probably did not make a wise choice at clue 2! How many times can you count the letter 'S' at the top of the sign? This number is X in the coordinates for your next clue: N 50° 56.029' W 1° 24.(X-4)(X-2)(X-5)'.
Be warned, you are now entering a place that is well used by the public, and so the Black Tower servants could easily be posing as that jogger, or that man on a bike, or that woman with a push chair.
Clue 4
You have arrived at a junction with two ways to choose, so which way now?
Clue: Admiral Nelson's elbow was shattered by a musket ball during the Battle of Cape St. Vincent, and his arm was removed by a surgeon without any anesthetics. But which arm was it?
Continue on your chosen path and then go straight ahead at the crossroads. After that keep walking along the path until you get to the next junction, and just enjoy the scenery. Do make sure you are not being followed though, the surroundings may be pleasant, but the Black Tower servants are not.
Clue 5
There are three paths to choose from here. The clue for you is: Which of Captain Hook's hands did Tick Tock Croc NOT eat? Take the correct path and enjoy a leisurely stroll until you reach another junction.
A word of warning though, chose carefully or you may end up in this long forgotten area of the Common:

Clue 6
Hopefully you have been making the correct turns! If so you should be approaching the crossroads shown below:

The man with a dog (circled) may appear like just another man with a dog, but he is in fact a known Black Tower servant and may well be patrolling the area.
Once you are sure that is safe to proceed you are face with three choices: left, right, or straight ahead.
Your clue is in the title of a 2014 Nicholas Cage movie. Make your choice and then walk a short distance to the next choice: straight ahead or sharp right.
Clue 7
As just mentioned you have a choice here: straight ahead, or sharp right? The photo below is your clue.

If you chose straight ahead, let X = 6 . If you chose sharp right, let X = 8
A clue awaits you at N 50° 55.8(X+1)2' W 1° 25.0(8-2)X'
Clue 8
How many types of dragonflies and damselflies can be found here? Use this number as X.
Now, whilst keeping a watchful eye, proceed to the next clue at N 50° 55.(X halved)(80+X)' W 1° 25.(X-8)0'
Clue 9
Look for a sign. According to the sign, how many places are straight ahead? Double this number and use it as X.
How many letters are in the place to the right, JUST BEFORE the roundabout? Add this number to the second digit of the first A road below this place. Use the answer as Y.
How many exits are there from the roundabout, in total? Use this number as Z.
You're getting close now! There's another clue to be found at N 50° 56.0ZX' W 1° 25.0Y'
Clue 10
What is the weight limit on the bridge? Add the two digits together and use this number as X.
You are nearly there, the artefact can be found at N 50° 56.0(X times 2, minus 2)' W 1° 24.(8 times X, minus 2)(X-12)'
The Artefact
Up to this point you have stayed on the paths (or at least you should have), to locate the artefact you will now need deviate from the roads and paths. But as you do, make sure those pesky Black Tower servants aren't lurking anywhere. Once you've found the artefact, you can find your way back by navigating to the original coordinates.