I wonder if I can make homemade discharge lamp (not hid). Maybe a glass tube with argon or other gases inside and wire electrodes, powered by those battery high voltage circuits. 

Well, as an equipment you need at least a decent vacuum pump and few small gas torches in your shop, then it becomes feasible.
But be ready for a lot of failures till you learn the skill (be patient, dont get deterred by those first failures), best would be to find some real neon sign shop and try to learn tbere, from those who have already mastered the art...
General glass work is a good start too - after all the lamp making is a precission glass work in the first place, so you may learn a lot even when learning to make glass pottery...

I remember when someone made own fluorescent tubes.

I remember when someone made own fluorescent tubes.

Do you have any picture of it? It will be interesting to see it. 

Do you have any picture of it? It will be interesting to see it. 

For example this: https://www.lighting-gallery.net/gallery/displayimage.php?pos=-149192

Using incandescent bulbs as fluorescent filaments. 

Using incandescent bulbs as fluorescent filaments. 

Now I noticed the incandescent lamps there! Cool tube 

What if I make a sulphur lamp?

You are really not aiming low...
Better start with something simple first, like cold cathode Neon tube or so.

You are really not aiming low...
Better start with something simple first, like cold cathode Neon tube or so.

Can helium from regular balloons works? I know that this helium is mixed with oxigen, for the users safety. I think helium will need higher voltage than just battery and ballast, maybe bug zappers will work.

I said Neon, not Helium.
First Helium is way more expensive, may damage some electronic (it tends to contaminate MEMS resonators used in RTCs in some modern phones or "smart watches"; fortunately HW wise it is reversible - it diffuses out and the the resonator recovers, but a question is, if the software does not crash during the time the RTC is acting).
Second the oxygen will be both afecting tye discharge, as well as corrode the electrodes (definitely in the amount that has to be used to make the gas safe even for i***ts attempting breathing it).

I think because you would need to get some specialised equipment (vacuum pump, set of gas torches,...), adding a small "bottles" of a signage grade Neon and Argon to the shopping list would be the easiest.
But seeing the questions, better to really look for some signage shop first and get first experience in the trade there. First you get advice (mainly in ithe initial stage, when "nothing will work for you"), second you will see if this is really a hobby you want to invest into (you would need a car garage sized room, well ventilated, with gas installation, some equipment,... - it is not that cheap). And when you decide it is the thing you want to do, the people from that shop may help you to set it up for as cheap as possible (a lot of the equipmend could be home made or restored from near-scrap condition, but you need some experience for that)...