Blame it on all the cowboy movies, but I've always enjoyed the sound of 'adios' as a farewell. I don't want to be burdened by the superstitious baggage that 'adios' brings with it. Can someone invent a short, sharp, euphonic Spanish protologism, neologism or nonce word/phrase that means 'thrive without gods'?
Well, to begin it should be made clear that, while adiós indeed comes from the phrase a Dios os encomiendo (I entrust you to God), like the phrase ojalá (from لو شاء الله meaning if God wills (it)), today it no longer is seen to have any religious significance. My atheist friends who are very particular about language1 have no problem using it. In that regards, it's much like using good-bye in English.
A simple one that you can use, but that I don't know it has much usage outside of Spain, is agur which derives from Euskera (Basque), but even that comes originally from Latin augurium so you aren't really removing yourself from "superstitious baggage".
Other farewells tend to be relatively context-sensitive, but you can often hear things like que te vaya bien (may things go well for you) which might additionally have other words attached, for example, que te/os/le/les2 vaya bien la vida (may/I hope life treat you/y'all well). But this isn't per se a farewell, and can be used in other, non-farewell contexts: if someone has told you in the conversation that they have a big test tomorrow, using it will be interpreted as wishing them well on the test, not generally. Additionally, you have to consider who you're saying it to and where you are.
Another potential option is cuídate/cuidate/cuidaos/cuídese/cuidense3 which means take care (of yourself/y'all's selves). But it too has the problem of multiple forms depending on who you're talking to and where you are.
The last group of farewells, which are probably the most common after adiós generally begin with hasta (until). For example, hasta mañana (lit. until tomorrow, that is, see you tomorrow) or hasta luego (until later). If you won't ever see them again, you can use the phrase hasta nunca (until never, that is, hope to never see you again), but that has a very negative feel and certainly doesn't wish them well. You substitute te/os/lo/la/le/los/las/les veo instead of hasta (lit. I (will) see you/y'all), but it has so many different potential words based on where and who you're speaking to, if you're not actually a Spanish-speaker it might be best to avoid it.4
There is one final option that you could potentially use. It is short, and unrelated to deities. It is vale which derives the identical Latin phrase meaning stay(sing.) well. The problem (and honestly why it didn't come to me on the initial writing of this answer), is that in a number of countries —especially Spain— vale is indeed used as an interjection, but one meaning closer to the English word okay. When it is used in Modern Spanish with the meaning of farewell, it tends to be in a very different context: funerals. For example, vale, mi amigo would mean farewell (forever), my friend, and wouldn't be super appropriate in more, shall we say, cheery circumstances. It also has a number of other meanings, some of which could be confused in context: in the Dominican Republic vale can be a rural person, so telling some vale could be the same as calling them redneck or countrybumpkin to their face.
1. By this I mean my friends who are both particular about language and atheists (which are of course unrelated traits). By particular, I mean having a high tendency to use things like x or @ to neutralize gender, etc. The ones who are very concerned about the subconscious effects of language.
2. te (informal or intermediate formality, singular), os (informal and plural, Spain and Equatorial Guinea only), le (formal and singular), les (formal and plural, Spain and Equatorial Guinea; plural without distinction to formality everywhere else)
3. in order we have first the informal (or intermediate formal in countries with three-way distinctions like Chile), informal singular (for countries with vos like Argentina), informal plural (for Spain and Equatorial Guinea; very formal if used in singular), formal singular, and plural (formal in Spain and Equatorial Guinea, no distinction elsewhere).
4. Suffice it to say, we have informal and intermediate formality forms (te, singular, os plural in Spain and Equatorial Guinea), formal forms in non-leísta or leísta de cortesía zones (lo singular masculine, la singular feminine, los masculine plural, las feminine plural) formal forms in leísta de cortesía zones (le singular without gender distinction, les plural without gender distinction), or formal forms in standard leísta zones (le for masculine singular, la for femoinine singular, les for masculine plural, las for feminine plural)