Christianity brings three significant advantages to the ambitious early medieval monarch: community, symbolism and bureaucracy.
Augustine of Hippo's City of God formed the basis of post-Roman Christian theology, positing 'Christendom' as the successor to the Roman Empire. Being a Christian ruler meant being a member of a family of rulers, unified through faith despite political differences. Christianity provided a starting point for diplomacy and friendly relations between rulers: As Charlemagne and Offa sought to bring Francia and Mercia closer together in the 8th Century, Charlemagne addresses correspondence to Offa with...
Charles, by the grace of God king of the Franks and Lombards, patrician of the Romans, to his esteemed and dearest brother Offa, king of the Mercians, sendeth health; First, we give thanks to God Almighty for the purity of the Catholic faith, which we find laudably expressed in your letters.
Being part of the Christian community ties into the second point: symbolism. I wrote here about the importance of Romanitas to Early Medieval kingship and the projection of power and authority. Actively patronising the Church was a clear and ongoing way to clearly showcase Romanitas in the Augustinian style. Church buildings in stone were also an open display of a leader's ability to muster the knowledge and resources to undertake such construction.
The most immediately practical benefit came through the bureaucracy of the Church. Catholicism brought literacy, numeracy, laws, record keeping and a clearly defined heirarchy with an easy reproducable format, which allowed a newly-Christian monarch to rapidly expand structures of goverment across their territories. The establishment of a bureaucracy allows kingship to expand beyond the territory that could be immediately controlled by the physical presence of the monarch and the bounds of personal loyalty tied with feasting and gifting, and allow for laws, charters, treaties and reports to establish control over an ever-wider area.
I've written on this topic several times, here are some of my answers