How much would you spend on this project i.e. price of sentimentality?
Be careful, any rework or mods could affect safety and your insurance coverage.
(1) You might be able to find direct replacement for the neon tubes but most common neons like NE-2/2H are much shorter and probably a lot less bright.
(2) You could use standard NE-2 (or 2H) type neons, put two or three in a row, but you'd wire them in parallel and likely need to change the dropping resistors; a bit messy.
(3) You could replace with incandescent bulbs (removing dropping resistors) but they will run hotter, which might not suit the plastic body; and they need to be the right voltage. Also, incans have a habit of burning out which means opening it up to replace them.
(4) Small (candelabra) LED filament-type bulbs which run directly off line are available; again remove dropping resistors. You might find others with different shape i.e. cylindrical but what I've seen tends to pricey.
Look for 2200K colour temp (CCT) which would somewhat mimic the neon colour (assuming it's orange); but most are typically 2700K or higher. Common C7 bulb may or may not fit inside the star; check with available (incan) Xmas/nightlight bulb.
Globe Electric has some which are 0.5W, 35 lumens, 2700K; see them at HD but a bit pricey though.
(5) Small candelabra non-filament LED bulbs are available. I use them in nightlights. Some are pretty impressive for 0.4W. Light dispersion on some is very good, for nearly point-sources.
https://www.feit.com/product/4w-equivalent-2700k-led-night-light/
(6) You might work with LED filaments directly but I don't recommend it. Typical one in ac bulb requires about 70v or so but needs proper current limiting circuit (which is built into base of bulbs). You need to know what you're doing.
Dave