Hey, a bit of goodwill :) Sorry for the arrogance tho, i think it's slightly misguided : Instead of trying to rebuild their reputation, they could try reforming their country to rebuild their futur.
There are several actions they could take for example. But it might be, well, unpopular to say so :
Sadly, to progress technologically a society must also progress a bit socially, and currently we have only one known path in occident known to work. If they want our thunder, they'll have to walk the same path we did at least partly : That mean becoming partly secular. What i mean by that, is for example they'll have to become more tolerant of different religion (in particular other branchs of islam, like the shias, supported by their rivals), accepting that people can have no religion, or not be interested in islam (despite their parents being muslim). I understand it's unpopular because it means partly giving in to Iran or foreign powers from their view. But, lessening religion is going to be important if they want their society to progress.
Finding a way to... calm down on salafism. Radical Salafism is sadly partly incompatible with a number of western values that made occident the superpower it is. In understand they don't give a crap about it; but money alone can't fix a society as they found it out with the oil money. If people want a society working like ours, they'll need to adopt some of our values.
It's going to sound like a ridiculous insult and agenda pushing, but that also mean they'll need to teach their population that women aren't inferior. You can't have a working society without half your society working on equal footing.
Stop with the mass killings. I understand the US do it too, but it's not a good example. The campaigns of beheading, disappearance and torture are more than highly impopular in occident : They're holding society back.
Economically, they need to "diversify" in an emergency. What i mean by that is they need companies doing something else than oil, and they'll need it soon. Like 20y horizon. For example, they could use a service sector, high tech research, stuff like biotech or computer technology, industry, etc.
Long term projection regarding survival with their neighborhoods. It's obvious the Saudi military has trouble, so they can't assume they can destroy every neighbors that's going to annoy them. That mean they'll probably need to compose with Iran like they cooperated with Turkey or the country arabs tend to call "the great satan", i mean Israel.
If they want to reform faster their country, they'll need to open it too , to "heavy hitters" country that have leverage like the USA. Currently they have good relations with the last one, but not with it's population nor with Europe; in big part due to their image of a "radical salafist country that bread terrorism". Sadly, once again to restore good relations with other powers, they'll have to use bracks on Salafism.
About that, it means not spreading their brand of Islam to other countries using money. Radical preachers have been a serious problem in Europe, but other non European countries might end up very, very hostile if somebody else was radicalized first in a saudi funded mosque. I understand Salafist Islam is the core of Saudi culture, but it will simply cause huge problem in other non saudi countries too. Trying to export it might My point is for example if the US under trump (very hostile to it) were to have another attack like it happened before, and it was by salafist from a Saudi-paid mosque, they could take it as an act of war. On the countrary, if the Saudi helped deradicalize people, by preaching a more mild brand of islam, it might exonerate them entirely from whatever salafist issue crop.
I also understand that the house of Saud's power rely largely on the cooperation of salafist clerics in the kingdom at home. That is , however, a problem, to which we have no solution to offer and that they will have to face inevitably : Those clerics are holding them back, holding back their entire country and society, and damaging the arab world. They will have to find a way to convince them to become... more open , or face the risk of another Arab spring in several decades when the next generation of the population will want western-like wealth and society. Or to find a way to otherwise weaken the clerics' grip on their country without destabilizing it. Large campaigns of propaganda in school might be helpfull to help society become a bit less conservative and more aligned with the values of the royal family itself.
This is simply a few suggestions; but my point is PR campaigns won't really fool our population, no matter how hard we try. If Saudi Arabia want a better rep as a country and better ties with the rest of the world, they'll have to actually do a few deep social/countrywide reforms. Partial "Liberalization" on their own terms. The problem part being of course how to do them not to fast to avoid destabilization/unrest.