trying something.
I noticed that the mysql dbases were different between two sites i had. one site was running mysql 4.1 and the other was running 5.0. the 5.0 was working easily.
I will try to update the site to 5.0 and will see if it works.

its not mysql 5.0, still did not work. i will try the other way listed above when i have more time.

I'm also seeing this problem.  I can install any version up to 1.14 without running into this problem, but 1.14.0 through 1.14.3 produce this error.  Installs okay on my local test server.  Tried making the suggested PHP.INI changes (mysql.connect_timeout = 60, max_execution_time = 600, memory_limit = 36M) in my site's root (which does seem to override the Godaddy globals) but they made no difference.  I'm working with Godaddy on this and will post anything of interest.

My live site is on Godaddy:
PHP built On:  Linux p3slh113.shr.phx3.secureserver.net 2.4.21-53.ELsmp #1 SMP Wed Nov 14 03:54:12 EST 2007 i686 
Database Version:  4.0.27-max-log 
PHP Version:  4.3.11 
Web Server:  Apache 
WebServer to PHP interface:  cgi-fcgi 
Joomla! Version:  Joomla! 1.0.15 Stable [ Daytime ] 22 February 2021 23:00 UTC 

My test server:
PHP built On:  Linux lamp 2.6.20-16-server #2 SMP Sun Sep 23 19:57:25 UTC 2007 i686 
Database Version:  5.0.38-Ubuntu_0ubuntu1.1-log 
PHP Version:  5.2.1 
Web Server:  Apache/2.2.3 (Ubuntu) mod_python/3.2.10 Python/2.5.1 PHP/5.2.1 mod_ssl/2.2.3 OpenSSL/0.9.8c mod_perl/2.0.2 Perl/v5.8.8 
WebServer to PHP interface:  apache2handler 
Joomla! Version:  Joomla! 1.0.15 Stable [ Daytime ] 22 February 2021 23:00 UTC 

UPDATE: GoDaddy says they can't help with application problems, which they've clearly decided is the case here.  I asked them about MySQL settings, since other people on these forums are speculating that the problem is there.  Meanwhile, it seems to me that if I can install any version prior to 1.14.x with no problems, but nothing from 1.14.0 onward installs without this error, something must have changed in 1.14.0 that is causing this.  I think it would be great if an RSGallery2 developer would dig into this a bit to see if maybe a change that appeared in 1.14.0 is causing this and could perhaps be reworked.  Others have speculated that 1.14.0 is making some invalid assumptions about Joomla (i.e. that it's 1.5, not 1.0.x).  Failing that, some specific recommendations regarding settings would be helpful.  Thanks!

UPDATE: Godaddy says there are no MySQL settings that can be set by the user; they are all global.  If it turns out I need to change a MySQL setting to avoid this problem I'll be looking at changing to a different hosting provider.

UPDATE: I seem to have resolved this by making the following changes to my Godaddy root PHP.INI:
mysql.connect_timeout = 120 (previously tried 60)
max_execution_time = 600 (did this earlier to no effect)
memory_limit = 36M (did this earlier to no effect)
max_input_time = 120 (hadn't tried this before)

Not sure what fixed it but I'm happy anyway.