Your description is fairly common and the fans are easy to service. I do this to every TiVo that comes over my workbench for repair or modding:

Firstly, unscrew it from the chassis. You'll need a Torx-15 screwdriver - also handy for hard disk removal from brackets if you ever need to, and a Torx-10 for the case. You probably have those but handy info for anyone else reading this.

Peel off the sticker in the centre of the fan and under that you will find a rubber plug. Carefully remove it with a sharp knife and you will see the tip of the axle the fan blade assembly spins on.

Grab some WD40, RP7 or whatever your choice of spray lubricant and give the whole axle and bearing a drink down the centre - no need to remove the axle just squirt away in the hole where the plug was.

Then put the rubber cap back on. Don't worry about the label throw it away.

That will keep things going for a long time.

EXTRA INFO (and what I do): Technically you CAN remove the plastic retaining circlip on the axle and remove the fan assembly but for the first timer it may be tedious and end up breaking or losing it. I also use white lithium grease but I do a LOT of fans, so for the one off while some argue WD40 is not the best choice it is the easiest and most widely available to most and does the job.