The answer is going to be inevitably complicated due to the multi-ethnic nature of the empire. I've no idea as to Hebrew and Armenian names, for example.

In Greek, the largest single corpus, the tendency is for either Roman names (Konstantinos, Romanos etc), biblical names (Ioannes, Michael) and Greek names which come in two stripes i) names to do with god (Theodoros etc) and concepts (Irene 'peace') or some times ii) classicising names (like Sophocles) though the latter are rarer.

I'm not sure how you'd count names from other languages into Greek like e.g Vardas from Armenian. If you want just raw data be sure to check out the the Lexicon of Greek Personal Names (LGPN) which gives you lots and lots.

I don't think that's a question that can be answered; but I'm going to give you a very specific, very limited answer and I hope it's close to what you're looking for.

These are the names of the paroikoi included in the estates granted by Emperor Michael VII to his cousin, Grand Domesticos Andronikos Doukas, in 1073. ( from Acta and Diplomata Graeca Medii Aevi Vol.6 )

So, the most popular names in the villages of Varys, Olynthos, Gamma, Vervoulidion, and Galaidai, in the Thrakesion theme, in the late 11th century are:

Male: 14 Georgios, 12 Ioannes, 5 Michael, 5 Leo, 5 Konstantinos, 4 Kyriakos, 3 Vasileios, 2 Moschos, 2 Nikephoros, 2 Strategos, Theodoulos, Vardas, Hasan, Nikolaos, Samonas, Photeinos, Stephanos, Agapetos, Petros.

Female: 12 Anna, 11 Eirene, 9 Maria, 2 Theophano, 2 Kale, Theocharistene, Kalla, Theopempte, Elisavet, Elene, Adriani, Agatha, Theriane, Sophia.