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I want to make a simple database for my backend server but I don't need something the size of mongo. I basically just want to track if the weather for a city has recently been sent to a client and cache it for a short time. I don't expect high traffic initially, and I may just store the json data in files and look at the timestamp, but would prefer an embedded database, since I will just have one server. I don't want to use H2 as I want the data saved. Kodein-DB states it shouldn't be used on the server. What else would work well?

H2 has both in memory and server options <https://www.h2database.com/html/features.html#embedded_databases|https://www.h2database.com/html/features.html#embedded_databases>

H2, hsql, or sqlite are all good options here

Rocksdb is also good if you want a more raw kv store and not full sql engine

Depending on the size of the cache - just caching in memory can also be a viable option

if you want caching, then perhaps check caching solutions?
hazelcast, redis, ....

Does a server need to know its clients? The client could just ask for the data when it feels like it, and be told "it's the same" or "here's the new data". This is http.