It looks to me as if you need Sane-hp_rts88xx file installed in /etc/sane.d/. This is a copy of part of the man page.
The sane-hp_rts88xx implements a SANE (Scanner Access Now Easy) backend that provides access to RTS8891
chipset based scanners. This backend will try to support the following models:
MANUFACTURER: MODEL: USB ID:
--------------- -------------- ---------
Hewlett-Packard Scanjet 4400c 03F0-0705
Hewlett-Packard Scanjet 4470c 03F0-0805
Hewlett-Packard Scanjet 3500C 03F0-2205
This man page also talks about if you get a 'blank' page and what to do with it. From what I could figure out, there may be different chips in some models of the 3500c.
If you run sane-find-scanner, it should find the scanner and report the chip in your device. Try that, and have a look at 'man sane-hp_88xx'.
I would suggest you have a look at /var/log/syslog to see if there is anything there listed that may help point at the problem.