How accurate is Tom Holland's claim in his book Dominion that most of our current values, including human rights and secularism, are based on Christianity?

The question of whether Christianity is responsible for basically all modern ethics is too broad in geography and chronology for one historian to answer. As OP has not read Holland's book here, and neither have I, let's put it aside. I can only speak for the Middle Ages and what went into medieval ethical theories, but even in a time that appears to have been under near total Christian dominance, it was not the only game in town. The role of classical philosophy in forming attitudes is increasingly investigated and accepted by current researchers, including myself. In other words, Christianity was very powerful, but still one ingredient in a moral stew. I also wonder how closely those who make such arguments have actually read some of the key pre-Christian texts or considered their impact even when Christianity dominated. I could go into chivalric ideology and how it often stood in opposition to what the papacy was preaching, but instead I'm going to specifically use De Clementia (On Mercy/Clemency) by Seneca to illustrate that the notion of pity as largely foreign in pre-Christian society is a common misconception.

Let's start with what appears true: that classical ethics had little time for pity. From Seneca's De Clementia, written to Emperor Nero:

At this point it is relevant to ask what pity is. Most people, after all, praise it as a virtue and call a man who shows pity a good man... The fact is, [pity] is the defect of a weak mind that succumbs to the sight of other people's miseries. Consequently, it is most familiar to the worst kind of person. There are elderly women and silly females who are so affected by the tears of the nastiest criminals that they would break open the prison if they could. Pity focuses on the situation, not its cause, whereas clemency sides with reason.

And if we looked at this and similar classical texts without a closer reading or considering the wider context, then we would indeed conclude that classical ethics are harsh and hooray Christianity for giving us pity, and from that pity rights and dignity etc. Except Seneca straight up tells us that most people have a concept of pity, and at this point Christianity was nothing more than a weird little cult with almost no power or influence. Seneca himself is sensitive to the idea of Stoics as harsh. He continues:

I realise that among the ill-informed the Stoic school has a negative reputation for being excessively harsh and least likely to give good advice... it is criticised for saying that the wise man does not show pity or forgiveness. These ideas, if they were stated in the abstract, are horrible, because they would appear to leave human error no hope, but to refer all failures for punishment. But if this is the case, what is this knowledge that tells us to unlearn our humanity and that closes the harbour that is our best security against Fortune by denying assistance? But in fact, no school is more kind or lenient, none is more philanthropic or more concerned about the common good - so that it is its objective to be useful, to be helpful, and to consider not only its won interest but that of all communities and individuals.

Seneca basically going "Listen here you little shits, you've misunderstood the last paragraph so let's go over it again" is one hell of a power move, and one that makes the point I also want to make. Abstracted, classical ethics suck. Put in context, classical ethics has pity and dignity and rights and all those nice things. Seneca refers to the cosmopolitanism of Stoic philosophy, and uses it throughout De Clementia to explain why everyone, regardless of their wealth or social status, should be afforded clemency if they fuck up and helped if they are unable to help themselves. He believes in a collective humanity, and a collective moral responsibility to look after each other to preserve human dignity and guard against the rise of cruelty. In this rather short text, Seneca refers to the idea of humanity and human dignity approximately 60 times. What he, and Stoics generally, object to is being moved in the moment by a base emotion. Seneca's message isn't one of "be merciless", it's one of "if you want to actually help, go home and think it over so that what you do will be maximally effective". As he says:

The wise man will not experience pity, but he will offer help and he will be of assistance, because he is born to benefit the community and the common good... He will offer his goodness in appropriate measure even to victims of disaster and to those who merit disapproval and to those who merit correction; much more readily he will come to the rescue of people who are truly afflicted and struggling through chance.

And what we have here is a semantic misunderstanding. What Seneca and other classical writers mean by "pity" is not the feeling of being moved to compassion, which is perhaps how many people would use it today. Instead, it is a sudden, gut wrench of misery toward the suffering of others. And many classical philosophers made the point that acting on that gut instinct was generally not an efficient way to help people. Seneca wanted the emperor, and the Roman government more generally, to help the poor and desperate. Which they did, through things like the grain dole and disaster relief efforts. Pre-Christian Romans certainly had a concept of "These people are suffering, and the state should help them."

Western Europe seems to have rediscovered De Celementia around 1090, with a couple of manuscripts dating from 1090 and 1093. In the 12th century, however, we find over 300 manuscripts of De Clementia, often produced in a combined edition with On Giving, another of Seneca's texts. They corresponded to the medieval chivalric virtues of clemency and largesse, so they became popular. According to Gerald of Wales, King Henry II of England was particularly keen on On Mercy and kept a copy at his bedside. This classical treatise was one of the most popular of the Middle Ages, read by counts, kings, and emperors from England to Bavaria. It was passed around tournament grounds and fuelled a thirst for Stoic philosophy, and Stoicism - or rather, the Christianity compatible, Stoicism-lite with all the metaphysics stripped out - clearly influenced knights. In the 1200s they traded texts by Stoics (or works falsely attributed to Stoics) at tournaments, held public readings of Seneca, and absorbed aspects of it into chivalric ideology. Even when Christianity dominated society, it was not the only game in town. As Seneca argues, it is simply human nature to feel pity and be moved to respect human dignity.

There is then the question of how much Christianity was shaped by pre-existing philosophers (it's not as if Jesus was the first guy to suggest being nice to each other) and where the buck stops, but that's a whole other thing.

Sources

Baraz, Daniel. "Seneca, Ethics, and the Body: The Treatment of Cruelty in Medieval Thought." Journal of the History of Ideas 59.2 (1998): 195-215.

Bartsch, Shadi, and Alessandro Schiesaro, eds. The Cambridge Companion to Seneca (Cambridge University Press, 2015)

Seneca, De Clementia trans. Susanna Braund (Oxford University Press, 2009)