Hi Vicki,
here's my three pennies worth!
Not many people have many camcorders to directly compare, so their views tend to be subjective.
Here's a link to a review site which you should find useful.
http://www.camcorderinfo.com/ratings.php
The highest ranked standard def camcorder is a JVC, at #25 in their rankings, and with the best "street" price coming in under your $500 budget.
Note that there are some high def camcorders which can also be had for under $500, but there are a couple of major problems with those.
First of all, you need a very powerful pc, preferably a quad core, to edit the video, and secondly, there are as yet no commonly available consumer video editing programs that can handle the footage without problems.
Not only that, but even if you can edit the video, it's presently uneconomic to produce a blu-ray disc from your project, so you have limited options as to what to do with your projects. Blu Ray burners still cost too much for most people, and the blank discs go for silly prices too. I'd hate to be making Blu Ray coasters...!
With standard definition, it's pretty simple - you can make up DVD's that you can pass on to family and friends.
Amongst the standard def camcorders, if you don't choose a Mini-DV model, they will record to mpeg2, either onto a hard drive or a Mini DVD disc. I would recommend strongly against buying a Mini DVD camcorder, however. At the high quality setting, you can only record 15 or 20 minutes of footage to a disc.
The hard drive camcorders are more compact, and they have the benefit of allowing you to transfer video to your pc faster than real time, in contrast to any tape-based camcorders. All non-Mini-DV camcorders record to a compressed format, however, which is less easily edited than Mini-DV.
There may also be a few flash memory standard definition camcorders, and they will be the smallest units available. Suitable high capacity flash memory cards are now pretty affordable, and certainly more convenient than tapes, and possibly more so than hard drives even.
You should read a few reviews to see which model does best in poor light - there are big differences between models from different manufacturers in that respect, and all the reviews on camcorderinfo.com show screen shots when used in low light which are very useful for comparative purposes.
To be honest, in your position I might consider a Canon HV30. It's a little above your budget, but you might see a good offer on one. It is a high definition camcorder that records in the HDV format, recording to a Mini-DV tape. That's the main drawback - you still have a tape based camcorder. But on the up side, you can re-use your old Mini-DV tapes if you like, if they're good quality, though many people never over-write their Mini-DV tapes and use them as an archive.
Although compared with the hard drive and flash memory HD camcorders the HV30 is large, compared with earlier generations of Mini-DV camcorders, and your TRV350, it will probably seem quite small. It's a popular model so you should be able to go down to your local big box store and play around with one, and compare it to the other camcorders while you're at it.
The great thing about the HV30 is that your existing pc will be fast enough to edit the HDV footage - but you even have an option to shoot in standard definition Mini-DV format too. Video Studio has no problems editing HDV either.
With HDV, you still have the problem of how to distribute your high def projects. Your options include keeping your output on a hard drive for playback on your pc or thru to an HDTV, or burning a "hybrid" HD disc onto a standard DVD blank disc for playback in a suitable Blu-Ray player or a Sony Playstation 3 hooked up to your HDTV. The PS3 happens to be a very versatile playback device, as well as a very nice boy-toy!
The HV30 also just happens to still top the camcorderinfo.com rankings, despite the introduction of lots of newer AVCHD camcorders. Bear in mind also that camcorderinfo.com doesn't really give much weight to the "editability" of video in their reviews, so if that's an important factor for you, you might rank it even further above the other camcorders.
The HV30 has very good low light performance too.
I think when you get a replacement for your trusty old TRV350, if you get a standard def camcorder it might be noticeably smaller, and perhaps have slightly better low light performance, but the image quality will probably be much the same as the TRV350. By contrast, the HDV footage from the HV30 will look fantastic. If that's a recent snap of your little girl, I would go for the HV30 just to make sure you get the best footage of her growing up. I'd shoot in HDV, archive all the tapes, and maybe produce some standard def DVD's from the footage for current distribution to family and friends but also having options to make up high def projects too. Best of both worlds really.
If you did buy the HV30, you might have to be prepared to see less of your husband! It has so many features that he might lose himself in it....