Sorry about the choice of words. A weak signal normally results in a low connection speed and episodes of drop out or fading. For gaming computers where performance counts, a weak signal is not suitable. However, for browsing the internet, a weak signal may be usable - if you can tolerate a low connection speed.
Boosting the wifi signal is a matter of the antenna and its location. There is an active thread on this board where they are discussing boosting the wifi signal by constructing a antenna reflector for the router. However, antennas are a 2-way proposition. Improvements are needed at both ends.
Wireless bridge and range extender both describe the method where a wireless network is created between two routers. Both routers must have this capability. The fios router probably has the capability. Your old router probably never gave a thought to the usefulness of this feature.
For the price of an Ethernet cable and the hassle to drill holes you can connect the two routers.
If you are ok with spending money, then consider extending the network. Your wireless laptop makes an excellent measuring tool for signal strenth. The laptop would be useful to determine if this could be accomplished point-to-point, or if an additional range extender is needed between both ends.