IMHO trying to improve the HSF result at Jutland is like trying to wank the Japanese position in WWII. So many strokes of luck and opposition errors are made IOTL that it is increasingly difficult not to resort to hoping that the lucky streak continues.
If we examine the errors made by the HSF I really can only see three, and two of those if remedied instead of increasing the chance of a German victory instead lessen the chance of a British victory - so a slight improvement: -
1. Leave II Battle Squadron in port. The pre-dreadnoughts reduce the overall speed of the HSF. I don't follow the idea that Scheer came over all sentimental but instead realised that if he did run into the entire Grand Fleet he'd be seriously outgunned, and might need those extra old rifles in an emergency. And, to be fair, the pre-dreads weren't too much of a drag in the important points of the battle, and did provide a precious shield when fending off Beatty's battlecruisers in the evening. Was that worth the loss of the
Pommern and her entire crew?
2. Why did Hipper (IMHO his only real error in the battle) or Scheer not wonder why Beatty was retreating pretty much due north - and even turned to the east at the end of the Run to the North? Retreat was NNW or further points to port. There must be something lurking out there. If they call off the chase they would almost certainly be castigated for not pressing home on an enemy then flying, but in hindsight a partial victory of 2 BC's sunk against damage only to the 1st Scouting Group would be a good return. Does the loss of
Invincible and 3 obsolete AC's balance the
Luzow,
Pommern, the light cruisers and the damaged remainder of the HSF?
3. Scheer's second turn back straight into Jellicoe's fumbling embrace. IMHO a huge error made in the heat & confusion of battle. If he runs for the Amrun Bank he could make cover of the minefields before Jellicoe figures out what's happened in time to pursue. (OK the pre-dreads might be an issue here!) This time you pretty much lose
Lutzow in exchange for
Invincible ,
Defence &
Warrior but do avoid a lot of heavy damage.
In return consider the numerous British errors that could be avoided: -
a) Beatty's performance... well, possibly not. Have a competent if less dashing commander of the BCF - Hood perhaps.
b) Effective pre-battle planning between Beatty & Evan-Thomas would bring 5th BS into action against Hipper much sooner - more potential damage to the enemy which might save the
Queen Mary at least.
c) Better communications from Beatty - & I don't just mean dropping Seymour overboard! Setting aside the fiasco with 5th BS there is the lack of information passed onto Jellicoe.
d) Improved shooting by the BCF. Hood's 3rd BCS managed to shoot just fine (
Invincible gets a credit in the eventual foundering of
Lutzow) and the reason they were at Scapa Flow was because there were better practice shoot facilities there than on the Forth. Surely that few weeks can't have made that much difference? Might not sink any of Hipper's ships but more damage may put guns out of action that damage / sink RN warships.
e) Cordite handling - seems that it wasn't just Beatty that condoned (ordered?) this unsafe practice as fingers were already being pointed at Jellicoe - and Callaghan before him. There is a school of thought that it was "easier" to blame the practice than to admit the BC's were underarmoured. It probably won't save
Indefatigable (reports give the opinion she was sinking by the stern before the explosion forwards) but might save
Queen Mary &/or
Invincible. And that not only makes an immediate difference in ships & men lost, but again more guns firing at the enemy should result in ore damage = less heavy metal being flung at the RN = even less damage in a geometric progression. Of course, a downside is a reduced weight of fire = less damage to the HSF = potential for them to actually improve the number of hits recorded, but at least the RN's crews have a much-reduced chance of immolation.
f) Better British shells. Jellicoe was well aware of his own ships' comparative lack of protection but he was also aware of the number of dud shells picked up in quality control testing. Sadly there wasn't the time to restock the Grand Fleet's magazines as improved shell production hadn't come on line in sufficient time. I understand that British shells at that time were designed to burst through armour with the power of the charge whereas German shells were made to pierce armour then explode. I'm ready to be corrected on this point.

I'm not talking Greenboys here but again an improvement in the quality of British shells would have an incremental difference at Jutland. Again more damage on the enemy is a two-way win - his ships not only suffer more but are able to hand back less.
None of these would be considered luck - there's no golden BB. The net result might be one or two German capital ships sunk or heavily damaged (either through better British shooting / shells or possibly hit by a reprieved RN BC) and one or two RN ships & crews saved. Even then just because
Queen Mary or
Invincible don't explode where & when they did IOTL doesn't mean they can't be sunk later on, or another ship suffer the same fate in their place - that's the problem with butterflies. But it sure is easier to come up with realistic scenarios where the RN performance is improved on OTL than to increase the HSF's prospects.