I don't know much about that radio but it sounds like a bad or intermittent solder joint somewhere.
If the radio has a hardware light switch (doesn't look like it does from the picture), I'd check it. If the light switch is in conjunction with the power switch I'd check that. If the lights seem to be firmware operated, I'd check the solder connections that supply the power to the lights.
But if it's surface mount, be careful. Surface mount is easy to mess up. Trust me, I know (I almost wrecked the headphone jack of a PRD5 'cause the solder spots were all surface mount).
Might be worth it to have a shop look at it if you look inside and see everything's surface mount... or live with the intermittent lights.
Sounds like it's a good radio (with some useful features) from reading some reviews. One review (
Ten-Tec RX-350 DSP HF Receiver Review - N9EWO -) mentioned some bad LCD's in later production runs but the guy may not have been referring to backlighting. The EHam reviews are positive.
I'd just take a look inside and see if the problem is something simple, like a loose cable to the lights or large solder spots that look easy to touch up with an iron. Otherwise, live with it or talk to a shop.... I'm sure others here who know the radio better will have a lot better info.
Good luck and Good DX with it either way.