Is it advisable to replace the internal HDD of an old Mac mini by a Solid State Drive?
Since the decoding of Flash videos primarily stresses CPU and depending on the codec and the video card the GPU an SSD doesn't help much (assumed you have enough RAM).

Another caveat is the internal HD Interface of the Mac mini 1.1/2.1: Serial ATA (1.5 Gb/s) which roughly equals to a max. throughput of 150 MB/s. Modern SSDs provide up to 600 MB/s / 500 MB/s (read / write). As a consequence you will lose up to 2/3 of the possible bandwidth by attaching the SSD to your internal interface.

So better save the money and get a more recent Mac.