I read a book by [name_f]Suzanne[/name_f] Weyn called Reincarnation several years ago that has this basic premise (nothing the same as yours, but it did go through several time periods). However, it had very popular time periods that were used.
Some ideas:
Pre-medieval bronze-age
[name_m]Stone[/name_m]-age / firstman times
Maybe ancient Mesopotamia?
Any time periods in [name_f]Africa[/name_f] and pre-Columbian Americas
Australia (pre-Europeans)
Pre-[name_m]Roman[/name_m] Germania
First Nations Inuit and Métis peoples of the Arctic pre-Europeans
Viking explorations of Greenland and Iceland
Medieval [name_f]Russia[/name_f] and Siberia
Ancient Celts
Anything really to do with the Portuguese
World War I (its just really kind of skipped over or you only get American flyboy or Romanov stories) and the African Americans that served
[name_u]Israel[/name_u]/Palestine post-WWII
[name_m]Pacific[/name_m] Islander stories
[name_f]India[/name_f] - any time period is underdone in literature
1930s Mexico & Central [name_u]America[/name_u]
Ottoman Empire
Post WWI [name_f]England[/name_f]
1980s [name_m]Germany[/name_m]
Pre-flood world
As for names, you could try to find names in the different cultures/time periods you pick that have the same meaning. I.e. maybe a character from [name_u]West[/name_u] [name_f]Africa[/name_f] is named Adebola, then they’re from Ancient Greece named [name_f]Coronis[/name_f], then [name_m]Justinian[/name_m]-age [name_u]Rome[/name_u] named [name_f]Corona[/name_f], and when they’re a Hebrew midwife while the Hebrew are retaking [name_m]Canaan[/name_m] post-[name_m]Exodus[/name_m] named [name_f]Atarah[/name_f], the a 1500s catholic priest in [name_f]England[/name_f] avoiding persecution named [name_m]Stephen[/name_m], and a Turkish soldier sent to the Crimean War named Aytaç, then a Tahitan teen in the 1950s named [name_f]Heirani[/name_f], then a 2010s beauty queen named [name_f]Tiara[/name_f]. All those names mean crown or relate to that meaning in some way. You could do the same with a lot of names. (star could be [name_f]Danica[/name_f], [name_f]Astraea[/name_f], [name_f]Estrella[/name_f], [name_f]Esther[/name_f], [name_u]Seren[/name_u], Nthanda, [name_u]Sterling[/name_u], [name_f]Tara[/name_f], Shihab, or Seong-[name_f]Jin[/name_f], beauty could be [name_m]Kalixto[/name_m], Calliphone, [name_f]Akemi[/name_f], Zurielle, [name_m]Hasan[/name_m], [name_m]Beauregard[/name_m], [name_f]Frida[/name_f], [name_f]Ingrid[/name_f], [name_m]Jamal[/name_m], or Marikit, tree could be [name_u]Rowan[/name_u], [name_f]Elowen[/name_f], [name_m]Alon[/name_m], [name_f]Melia[/name_f], [name_u]Lindon[/name_u], [name_m]Silvanus[/name_m], [name_m]Oren[/name_m], Yasan, [name_m]Itsuki[/name_m], [name_m]Garrick[/name_m], Björk [name_u]Forest[/name_u], or [name_f]Hadassah[/name_f], and ect…) I got most of these examples from the “themes” tool on behindthename.com, it is really helpful for stuff like that. But I do think that the idea of using the same initials throughout would work really well too, and would leave room for a lot more creativity and name choices.
Also, if you left them with personality traits that are similar (i.e. first they like to makeup songs, then they’re an actual singer/performer, then they write music, then they play an instrument, or they tend to gravitate to a certain colour or food-type, or they have a similar fear in each life, a birthmark that stays) little things like that could also help keep your characters distinguished from each other but they’d still be the same essence or soul they have always been, just in a different life and body.
Also, another thing, have you considered the idea that they aren’t always human in each life (i.e. a butterfly, tree, snowflake, tiger, raindrop, blade of grass, flower, sea monster, pet cat, ect…)? That could open up a few more possibilities if you were open to that idea.
I hope these ideas were of some help 