What kind of timer can I use to maintain 7 day timing on an old setup?
I am familiar with those systems:  They don't actually use relays, but solenoids.  They were made by several manufacturers including GE, Remcon, Bryant, and Sierra.  One instantly knows these systems by pressing an on/off button and hearing a 120 Hz buzzing somewhere (usually in the attic).

Each controlling rocker switch has two momentary contacts, one "on" and one "off".  Those are wired to separate coils around opposite ends of a common solenoid to either drive it in one direction or the other.  The presence of the solenoid (which has no spring) either pushes against or pulls away from a 120 volt switch to turn the circuit on or off.

With the most minor of maintenance, these systems continue to work reliably 50+ years later!

Any controller which interfaces with the system must do no more than a momentary connection.  Applying power to the solenoid continuously is "bad".  The existing switches can be left intact.  One of the big advantages of that style of system is that any number of switches can control the same circuit.

I am not aware of any controllers from the era.  Current aftermarket dealers, such as this, provide replacement switches, faceplates, solenoid assemblies, and transformers.

There are home automation systems which integrate with these, but they all seem to be pricey&mdash;$1,500+.

If you just need some security lights, maybe just use lamp plugged into a timer which plugs into an outlet?  (It's okay to ignore the existing system.)