You've misinterpreted the movement penalty for grappling and the movement restrictions for jumping. They limit different things.
From the PHB, page 195:
When you move, you can drag or carry the grappled creature with you, but your speed is halved, unless the creature is two or more sizes smaller than you.
So you're not halving your jump distance, you're halving your movement speed. And you have to use that for jumping (PHB page 182):
Either way, each foot you clear on the jump costs a foot of movement.
This means that, for a player whose normal movement speed is 30ft, while grappling he can move 15ft, and that's also the maximum jump height he can achieve. Not even the Jump spell or the Boots of Striding and Springing will let him jump further than that. The effects, however, do let him get around the Standing High Jump penalty.
To summarize, when looking into how high you can possibly jump while grappling, you make the calculations in a few steps:
- Determine your movement speed. This is also your maximum jump height (distance).
- Halve that height if you're grappling a creature that's not two or more sizes smaller than you.
- Determine your base jump distance. Use your total Strength to calculate this (including any bonuses to Strength from items, spells or potions).
- Multiply the base jump distance depending on spells an items you have that increase your jump distance (like the Jump spell).
- If you're making a Standing High Jump, halve the distance from step 4. Otherwise, subtract 10ft from the height in step 2.
- Take the lower number between the numbers that you get in steps 2 and 4 (after modifications in step 5). This number is how high you can jump (and how much movement you'll have to spend to jump the full height).
You can use this little tool I whipped up to test out various combinations.
If you find the maximum jump height (step 2) to be restricting you more than your jump distance achievable (step 4), you can use spells and items that increase your movement speed to propel you further. The Haste spell is good to consider. While it is a substantially higher level spell, it lets you use an extra action to Dash, quadrupling your speed in total. The added AC is a nice touch, too. You can even combine it with the Jump spell, as the latter does not require Concentration.
Using just the effects in your example, however, the highest you can jump is 15ft, using a Standing High Jump. If you somehow throw in a Dash, you can jump twice as much. Note that, if the PC's base Strength is already 18, the Belt does not help increase the jump height in this case.
Keep in mind, though, one litle bit about the High Jump rules:
In some circumstances, your DM might allow you to make a Strength (Athletics) check to jump higher than you normally can.
The rule enables you as a DM to let your player jump beyond his maximum movement speed (this bit of the rule is written directly after the bit about the movement speed restriction). When can that happen, how much of a bonus does the player get, what the DC is and what the possible consequences are (chance of Exhaustion, for example) - all of this is up to you.
A commentary on the Belt of Storm Giant Strength
I personally dislike the use of this item. First of all, it is a legendary item, suitable only for the most glorious of adventurers. Giving this item to a player character that's not very high in level (18 at least, I'd say) would seem overpowered, and frankly uninteresting. Even at that level it might not be as good as it sounds. Surely, it's a fantastic item, but it makes all the precious Ability points a grappler had spent on Strength a waste. On the other hand, having the belt in possession early and spending the points somewhere else (or getting feats), the grappler's main mechanic would now depend entirely on the belt. Losing the belt would thus make the character almost useless. While you might not plan on having it taken away from him in any encounter, the player might become paranoid about it nonetheless.
Alternative to using the belt, using a Haste spell (3rd lvl) and the Boots of Striding and Springing (uncommon item), the player can reach 21ft with a Strength of 18, and 24ft with a strength of 20 (assuming a base movement speed of 30ft, and making a Dash with the Haste extra action) by making a normal High jump (with a 10ft running start). Throw in a Jump spell (1st lvl) and the High jump goes up to 50ft. Using these three effects, the Belt would only give an improvement of 4ft with a Standing high jump.
If the Haste spell is not available, and using Dash is not an option, then a better combo would be to use Boots of Speed (rare item) and the Jump spell. Just those two bring the High jump up to 20ft, which might be enough. In this case, too, increasing Strength above 18 brings no benefits (though I'm sure it does so in the other aspects of grappling).
Seeing as all the Belts of Giant Strength do provide great bonuses to almost all aspects of grappling, I would award the legendary version at lvl 20 as a special Grappler capstone. Of course, not without effort. It's a nice excuse to lead your players into a perilous dungeon. I would also house-rule that the belt require a Strength of 20 to wear, just so all those Ability points don't feel like they're going to waste. If your player left his strength at 18, though, that might not be necessary.
At lower levels, a potion of Giant Strength (whichever tier you fancy) might be a suitable reward for a heroic deed, and since it's only temporary it wouldn't break anything (including fun).
There are no rules but 65 feet makes the most sense
Jump
You touch a creature. The creature's jump distance is tripled until the spell ends.
Remarkable Athlete
when you make a running long jump, the distance you can cover increases by a number of feet equal to your Strength modifier
Chronological order
The bonus to jump should follow the order in which the bonuses are actually applied in the game. In other words, since you apply the spell before you jump, your jumping distance is tripled first. Then, when you actually do a running long jump, the remarkable athlete bonus gets added onto that.
Thus, the math follows the order in which the bonuses are actually applied.
Constant versus situational bonus
Another way to reach the same conclusion is to think of the two effects in terms of how general they are. While under the effects of jump, your jump distance is constantly tripled. However, remarkable athlete only gives you the bonus to jump when you are actually making a running long jump. If you make a standing long jump you don't get any bonus at all from remarkable athlete, but jump still applies.
I would argue the constant general effect gets applied, and then the situational bonus gets added on top of that.
Either way you think about it, the math would work out to:
(20 x 3) + 5 = 65 feet
Note regarding jumping and speed
The distance you jump is indeed limited by your speed but with dashing and other means of speed increases the jump distances can be kept relevant as indicated in this conversation with Jeremy Crawford:
Q: Can you jump farther than your movement when using magic i.e spell
Jump & boots of striding and springing?
A: To be clear, things like the jump spell don't increase speed. You
can jump crazy far, but your speed caps it.
Q: Are you saying you can't jump farther than your speed even with
Jump spell or Boots of Striding and Springing?
A: Every foot jumped costs movement, so you can jump farther than your
current speed if you take the Dash action.
This is discussed more here Does a ring of jumping let you jump farther than your normal movement? and What happens when you run out of movement while jumping?
Best Answer
Jump does not let you bypass the 'Speed' restraint on jumping. But the ring is still useful
As to Jump (and the boots) not increasing your speed, here is some Developer commentary on the matter, specifically addressing these effects.
Source
So, no. Neither the Jump spell nor the Boots of Striding and Springing allow you to exceed your total Speed in distance jumped, though taking the Dash action allows you to move your full Speed a second time during the turn, effectively increasing your Speed by 2x for the round. This lets you actually get some use out of your massively boosted jumping distance. This would rather plainly imply that the Boots of Striding and Springing are generally superior to a Ring of Jumping.
Where this is not true is when you start combining magic items. For example, if you get your hands on a pair of Boots of Speed. Naturally, you can't wear two pairs of boots...but you can wear a pair of boots and a ring. This combination takes two bonus actions to fire up, but then you have a full minute of doubled running speed and tripled jumping distance, without having to take the Dash Action. And because you can get 10 minutes of boosted speed from the boots per day, you can do this 10 times a day. So you have all the mobility of someone Dashing around in Boots of Striding and Springing, but still have your Action free to use.
So, taken in isolation, it does appear that the Boots of Striding and Springing are generally superior to a Ring of Jumping. However, you have 10 fingers for ring-wearing, but can only wear one pair of boots. So the ring is better for combination work.