It could be the starter or ballast (transformer). Starters are dirt cheap. Start with that. 49 cents at most stores. FS-2 if it's an 18" fixture
As far as a ballast, Home Depot carries an IDENTICAL part number for most arcade games. It's a GE one for 14 to 20 watt fixtures. I've seen others by Sylvania that are a bit larger/heavier that will work with that range to a bit larger. I havn't tried those yet.
Price: $3.59 on clearance at HD for the GE. Or $2.89 at some places for the Sylvania.
Make sure you get one that works with a Size T8 (8/8th of an inch or 1" diameter), 18" and 15 watts. Don't go too high or you will risk fire or instant tube blowout.
Oh and *some* of those cheap all-in-one fixtures do not feature immediate (auto) start. You must hold a switch into a separate position which temporarily applies more volts to warm the filaments/tube up prior to starting before the ballast takes over to keep them running at a lower voltage.
Therefore they wont work in game units if you want it to turn on the marquee immediately without first having to hold down a button (behind the marquee

)