Not sure on your vehicle, I have a latter model es. But normally it is one of four things:
Most likely: Battery, Battery Wiring Cables & Wiring Harness needs to be replaced.
The clicking noise is the selenoid which is inside the starter which you already replaced.
A Lexus takes a ton of power to run because of all the extra accessories it has on it. Even if your lights and accesories are coming on your battery could not have enough voltage to turn the engine over.
Check how old your battery is- wipe off the top with a paper towel - it should have a sticker on it which will tell you how old the battery is. If it is old- over 5 years- could be the battery. You can have that checked for free at a big box auto parts store; or if you have the fancier jumper cables with the meter on it, it will tell you if it is putting out maximum power.
If you see corrosion, a white build up on your terminals or connectors:
You can buy a battery cleaner at an auto parts store for $5.00. Home remedy, use vinegar and baking soda mix together in a disposable plastic cup. Take an old hard bristle tooth hard or small steel brush is better, dip it in the vinegar & baking soda solution.
Clean your posts and and connectors off until you no longer see any white build up. Be extra careful because that white stuff is corrosive acid, can eat through your clothes, burn your skin, wear gloves or wash you hand thoroughly several (many) times afterward. If you do not disconnect the post wiring you can get a pretty good spark going if they happen to accidentally touch.
1) 80% of the time
- Most likely your battery harness cables have corrosion inside the wires leading to your battery which you can't see. If you look closely at the top of the sub wires (not the ones that attached to the batttery posts) and you see white powdery substance at the ends - you have to replace that battery wiring harness. You will have to cut & splice some wires together.
I could jump mine with a portable electronic power supply for several days when the clicking sound occurred, but eventually that did not work. It would not turn over and I could hear the (kiss of death) clicking noise. I don't think a battery to battery jump would be strong enough but it might be.
These are much less likely to have occurred:
2) You can check you starter fuses, relays and ignition fuses & relay. But Lexus does not have diagrams as to which fuses & relays correspond to which components on the back of the fuse panel, which is located inside the engine compartment. There are several relays - they are larger square looking ones. The fuses are easy to check, see & tell if the are broken inside - but the relays, I do not know how to test or tell if they are broken. Unlikely it would be your fuel pump - or electronic fuel injection relay or ignition relay.
A Toyota Camry is very similar, the es is based on. The es is a jazzed up Camry. It is a lot easier to find info on the Camry than a Lexus es 300.
Link to 1996 Toyota Camry fuse box:
Link to wiring diagram but not fuse box: http://www.commandocaralarms.com/wiring/1992-1996/lexus/es300/1253.html
3) Voltage regulator - I believe it is inside the alternator. But you should use a voltage measuring meter to test it before you fiddle with that. Buy it, check it, return it. Try to not get it to dirty.
4) Join Lexus Forums
- similar owners have better answers
5) If you can find a Haynes manual on your vehicle (on-line) or at your auto parts store - you could look in the back at the troubleshooting section. They will direct you to the simplest fixes first.
Because your vehicle is 19 years old it will be hard to find at an Auto Parts store.
On other older cars, I just read it at the store and don't buy it.
(I did get a suggestion off the net, which fixed another person's Lexus es that would not start, it failed to work for me. I tried to clean my fuel injectors when it would not start & the clicking noise was happening. Also, I changed my air filter which was filthy. The car ran a lot better but it did not fix the problem.)